Shift Name,Subject,Question Number,Question Text,Correct Option,question_id,chapter JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,1,"Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: (1) 628 (2) 812 (3) 526 (4) 784",4.0,1,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,other JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,2,"Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x(1) = 1$, then $x \left( \frac{1}{3} \right)$ is: (1) $\frac{1}{3} + e$ (2) $3 + e$ (3) $3 - e$ (4) $\frac{3}{2} + e$",3.0,2,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,other JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,ellipse JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,parabola JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,3,"Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: (1) 4 (2) 14 (3) 13 (4) 11",2.0,3,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,4,"The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$",1.0,4,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,properties-of-triangle JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,statistics JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,ellipse JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,5,"Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: (1) 19 (2) 24 (3) 21 (4) 22",4.0,5,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,properties-of-triangle JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,6,"Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: (1) 2 (2) 1 (3) $2\pi$ (4) $\pi$",2.0,6,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,parabola JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,7,"Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: (1) 7 (2) 4 (3) 3 (4) 5",3.0,7,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,8,"Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? (1) $\left( \frac{14}{5}, -3, \frac{22}{3} \right)$ (2) $\left( -\frac{5}{3}, -7, 1 \right)$ (3) $\left( 2, 3, \frac{1}{2} \right)$ (4) $\left( \frac{5}{3}, -1, \frac{1}{2} \right)$",1.0,8,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,ellipse JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,9,"Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_2 f(n)$ is equal to: (1) 2525 (2) 5220 (3) 2384 (4) 2406",1.0,9,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,statistics JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,10,"From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:",1.0,10,ellipse JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,logarithm JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,11,"Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: (1) \(24\pi^2\) (2) \(22\pi^2\) (3) \(31\pi^2\) (4) \(18\pi^2\)",2.0,11,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,12,"Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f'(0) = 4a\) and \(f\) satisfies \(f''(x) - 3af'(x) - f(x) = 0, a > 0\), then the area of the region \(R = \{(x, y) \mid 0 \leq y \leq f(ax), 0 \leq x \leq 2\}\) is: (1) \(e^2 - 1\) (2) \(e^2 + 1\) (3) \(e^4 + 1\) (4) \(e^4 - 1\)",1.0,12,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,ellipse JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,13,"The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: (1) \(3\pi + 8\) (2) \(6\pi - 16\) (3) \(3\pi - 8\) (4) \(6\pi - 8\)",2.0,13,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,14,"Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: (1) \(\frac{24}{5}\) (2) \(\frac{25}{9}\) (3) \(\frac{144}{5}\) (4) \(\frac{288}{5}\)",4.0,14,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,15,"If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: (1) \(0\) (2) \(\frac{4}{3}\) (3) \(1\) (4) \(\frac{1}{2}\)",2.0,15,properties-of-triangle JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,16,"A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: (1) \(51\) (2) \(64\) (3) \(32\) (4) \(48\)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,probability JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,17,"The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: (1) \(6\) (2) \(5\) (3) \(7\) (4) \(4\)",2.0,17,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,trigonometric-ratio-and-identites JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,18,"A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: (1) \(10\) (2) \(15\) (3) \(12\) (4) \(14\)",2.0,18,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,parabola JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,19,"Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: (1) \(36\) (2) \(31\) (3) \(37\) (4) \(29\)",2.0,19,circle JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 1),Mathematics,20,"Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: (1) 24 (2) 29 (3) 41 (4) 31",2.0,20,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,definite-integration JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,statistics JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,21,"Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________.",34.0,21,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,sequences-and-series JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,sets-and-relations JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,quadratic-equation-and-inequalities JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,functions JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,other JEE Main 2025 (22 Jan Shift 1),Mathematics,22,"If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________.",2035.0,22,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,indefinite-integrals JEE Main 2025 (22 Jan Shift 1),Mathematics,23,"Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.",16.0,23,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,differential-equations JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,binomial-theorem JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,parabola JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,differentiation JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,other JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,application-of-derivatives JEE Main 2025 (22 Jan Shift 1),Mathematics,24,"Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________.",34.0,24,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,matrices-and-determinants JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,3d-geometry JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,area-under-the-curves JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,complex-numbers JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,permutations-and-combinations JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,hyperbola JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,vector-algebra JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 1),Mathematics,25,"Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________.",216.0,25,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,1,"Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: (1) 253 (2) 154 (3) 125 (4) 157",4.0,1,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,other JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,2,"Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: (1) \( \sqrt{6} \) (2) \( -\sqrt{6} \) (3) \( -\sqrt{3} \) (4) \( \sqrt{3} \)",2.0,2,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,other JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,ellipse JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,parabola JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,3,"If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: (1) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (2) \( \frac{\sqrt{3} - 1}{2\sqrt{3} - 2} \) (3) \( \frac{\sqrt{3} - 1}{10(4+\sqrt{3})} \) (4) \( \frac{5 - \sqrt{3}}{14} \)",4.0,3,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,4,"Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: (1) 25 (2) 19 (3) 29 (4) 27",4.0,4,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,properties-of-triangle JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,statistics JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,ellipse JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,5,"Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: (1) 280 (2) 224 (3) 210 (4) 168",2.0,5,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,properties-of-triangle JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,6,"Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: (1) 2 (2) \( \frac{5}{2} \) (3) \( \frac{1}{2} \) (4) \( \frac{13}{6} \)",1.0,6,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,parabola JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,7,"If all the words with or without meaning made using all the letters of the word ""KANPUR"" are arranged in a dictionary, then the word at 440th position in this arrangement, is: (1) PRNAUK (2) PRKANU (3) PRKAUN (4) PRNAUK",3.0,7,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,8,"If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: (1) 2109 (2) 2129 (3) 2119 (4) 2139",4.0,8,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,ellipse JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,9,"Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is:",3.0,9,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,statistics JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,10,"Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: (1) 5 (2) 4 (3) 3 (4) 2",3.0,10,ellipse JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,logarithm JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,11,"Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. (1) 27 (2) 33 (3) 15 (4) 18",2.0,11,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,12,"The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: (1) 6 (2) 17 (3) 9 (4) 14",4.0,12,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,ellipse JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,13,"If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: (1) 58 (2) 46 (3) 37 (4) 72",1.0,13,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,14,"If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 195 (2) 179 (3) 186 (4) 174",3.0,14,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,15,"Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: (1) \( \sqrt{3} \) (2) \( 2\sqrt{2} \) (3) \( 2\sqrt{3} \) (4) \( 4\sqrt{2} \)",3.0,15,properties-of-triangle JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,16,"Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: (1) \( \sqrt{10} \) (2) \( 2\sqrt{3} \) (3) 2 (4) 3",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,probability JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,17,"Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: (1) 6 (2) 3 (3) 5 (4) 4",1.0,17,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,trigonometric-ratio-and-identites JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,18,"Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: (1) 3080 (2) 560 (3) 3410 (4) 440",4.0,18,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,parabola JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,19,"Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$ Then, the sum of all the elements in the range of $R$ is equal to: (1) $\frac{10}{9}$ (2) $\frac{5}{2}$ (3) $\frac{\sqrt{3}}{2}$ (4) $\frac{1}{3}$",4.0,19,circle JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 2),Mathematics,20,"If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then $(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: (1) 4 (2) $\frac{4}{3}$ (3) 3 (4) 2",4.0,20,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,definite-integration JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,statistics JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,21,"If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.",12.0,21,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,sequences-and-series JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,sets-and-relations JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,quadratic-equation-and-inequalities JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,functions JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,other JEE Main 2025 (29 Jan Shift 2),Mathematics,22,"Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________.",11132.0,22,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,indefinite-integrals JEE Main 2025 (29 Jan Shift 2),Mathematics,23,"If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________.",64.0,23,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,differential-equations JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,binomial-theorem JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,parabola JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,differentiation JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,other JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,application-of-derivatives JEE Main 2025 (29 Jan Shift 2),Mathematics,24,"Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________.",1328.0,24,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,matrices-and-determinants JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,3d-geometry JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,area-under-the-curves JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,complex-numbers JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,permutations-and-combinations JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,hyperbola JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,vector-algebra JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 2),Mathematics,25,"Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________.",10.0,25,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,1,"For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals: 1. 56 2. 132 3. 174 4. 280",4.0,1,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,other JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,2,"In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is: 1. 96 2. 144 3. 120 4. 72",2.0,2,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,other JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,ellipse JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,parabola JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,3,"Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals: 1. 5 2. 4 3. 3 4. 8",1.0,3,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,4,"Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to: 1. 148 2. 136 3. 144 4. 140",2.0,4,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,properties-of-triangle JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,statistics JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,ellipse JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,5,"If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is: 1. $\frac{4}{3} 2. \frac{7}{4} 3. \frac{5}{3} 4. \frac{3}{4}$",4.0,5,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,properties-of-triangle JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,6,"If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals: 1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ 3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$ 4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$",2.0,6,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,parabola JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,7,"The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is: 1. $\frac{8}{3}$ 2. $\frac{4}{3}$ 3. 8 4. $\frac{3}{2}$",1.0,7,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,8,"Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are: 1. 2 and 3 2. 2 and 1 3. 3 and 2 4. 1 and 3",1.0,8,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,ellipse JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,9,"Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:",4.0,9,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,statistics JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,10,"Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is: 1. 2 2. 1 3. 0 4. 3",3.0,10,ellipse JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,logarithm JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,11,"If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals: 1. \( e^{-1} \) 2. \( e^2 \) 3. \( e^4 \) 4. \( e^6 \)",4.0,11,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,12,"Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to: 1. 151 2. 139 3. 163 4. 127",1.0,12,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,ellipse JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,13,"Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to: 1. 6 2. 5 3. 8 4. 4",2.0,13,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,14,"The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is: 1. \( 4\sqrt{3} \) 2. \( 5\sqrt{2} \) 3. \( 4\sqrt{3} \) 4. \( 3\sqrt{5} \)",4.0,14,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,15,"The system of linear equations: \[ \begin{align*} x + y + 2z &= 6 \\ -2x + 3y + az &= a + 1 \\ 7a + 3b &= 0 \end{align*} \] If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to: 1. 16 2. 12 3. 22 4. 9",1.0,15,properties-of-triangle JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,16,"If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f(0) = 1 \), then \( f \left( \frac{1}{\sqrt{3}} \right) \) is equal to: 1. \( e^{\pi/12} \) 2. \( e^{\pi/4} \) 3. \( e^{\pi/3} \) 4. \( e^{\pi/6} \)",4.0,16,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,probability JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,17,"Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals: 1. 24 2. 25 3. 17 4. 27",2.0,17,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,trigonometric-ratio-and-identites JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,18,"The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is",3.0,18,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,parabola JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,19,"Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: (1) \( 1 + \frac{\pi}{2} \) (2) \( 1 + \frac{\pi}{3} \) (3) \( 1 + \frac{\pi}{6} \) (4) \( 1 + \frac{\pi}{4} \)",1.0,19,circle JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (22 Jan Shift 2),Mathematics,20,"Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7",3.0,20,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,definite-integration JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,statistics JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,21,"If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.",465.0,21,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,sequences-and-series JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,sets-and-relations JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,quadratic-equation-and-inequalities JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,functions JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,other JEE Main 2025 (22 Jan Shift 2),Mathematics,22,"Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.",3.0,22,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,indefinite-integrals JEE Main 2025 (22 Jan Shift 2),Mathematics,23,"Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.",145.0,23,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,differential-equations JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,binomial-theorem JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,parabola JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,differentiation JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,other JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,application-of-derivatives JEE Main 2025 (22 Jan Shift 2),Mathematics,24,"Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.",27.0,24,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,matrices-and-determinants JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,3d-geometry JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,area-under-the-curves JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,complex-numbers JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,permutations-and-combinations JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,hyperbola JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,vector-algebra JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,limits-continuity-and-differentiability JEE Main 2025 (22 Jan Shift 2),Mathematics,25,"Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.",28.0,25,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,1,"If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to - (1) $-1080$ - (2) $-1020$ - (3) $-1200$ - (4) $-120$",1.0,1,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,other JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,2,"One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is - (1) $\frac{3}{16}$ - (2) $\frac{1}{4}$ - (3) $\frac{3}{8}$ - (4) $\frac{5}{8}$",2.0,2,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,other JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,ellipse JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,parabola JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,3,"Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is - (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$ - (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$ - (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$ - (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$",4.0,3,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,4,"If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to - (1) $AB^{-1} + A^{-1}B$ - (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$ - (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$ - (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$",4.0,4,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,properties-of-triangle JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,statistics JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,ellipse JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,5,"Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is - (1) 52 - (2) 48 - (3) 44 - (4) 40",3.0,5,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,properties-of-triangle JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,6,"Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to - (1) 4 - (2) 32 - (3) 8 - (4) 16",3.0,6,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,parabola JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,7,"If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to - (1) 20 - (2) 5 - (3) 8 - (4) 10",4.0,7,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,8,"If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to",2.0,8,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,ellipse JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,9,"Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is \begin{align*} (1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\ (3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30} \end{align*}",4.0,9,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,statistics JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,10,"Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to \begin{align*} (1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\ (3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3} \end{align*}",2.0,10,ellipse JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,logarithm JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,11,"Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is \begin{align*} (1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\ (3) \ (0, \infty) & & \quad (4) \ \mathbb{R} \end{align*}",4.0,11,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,12,"\((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \) If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to \begin{align*} (1) \ 6 & & \quad (2) \ 10 \\ (3) \ 20 & & \quad (4) \ 12 \end{align*}",4.0,12,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,ellipse JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,13,"The number of words, which can be formed using all the letters of the word ""DAUGHTER"", so that all the vowels never come together, is \begin{align*} (1) \ 36000 & & \quad (2) \ 37000 \\ (3) \ 34000 & & \quad (4) \ 35000 \end{align*}",1.0,13,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,14,"Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is \begin{align*} (1) \ 10 & & \quad (2) \ 7 \\ (3) \ 8 & & \quad (4) \ 9 \end{align*}",2.0,14,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,15,"Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to \begin{align*} (1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\ (3) \ 2 & & \quad (4) \ 3 \end{align*}",4.0,15,properties-of-triangle JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,16,"The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is \begin{align*} (1) \ 2 & & \quad (2) \ \log_2 2 \\ (3) \ 1 & & \quad (4) \ e^2 \end{align*}",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,probability JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,17,"Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals: - (1) 50 - (2) 100 - (3) \( \frac{81}{25} \) - (4) \( \frac{121}{25} \)",2.0,17,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,trigonometric-ratio-and-identites JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,18,"The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is - (1) \( 2/3 \) - (2) 0 - (3) \( 3/2 \) - (4) 1",2.0,18,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,parabola JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,19,"Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I(37) - I(24) = \frac{1}{4} \left( \frac{1}{\beta x} - \frac{1}{c x} \right) \), \( b, c \in \mathbb{N} \), then \( 3(b + c) \) is equal to - (1) 22 - (2) 39 - (3) 40 - (4) 26",2.0,19,circle JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 1),Mathematics,20,"If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to - (1) \( x - \tan^{-1}\frac{4}{3} \) - (2) \( x + \tan^{-1}\frac{4}{5} \) - (3) \( x - \tan^{-1}\frac{5}{12} \) - (4) \( x + \tan^{-1}\frac{5}{12} \)",3.0,20,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,definite-integration JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,statistics JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,21,"Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to",19.0,21,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,sequences-and-series JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,sets-and-relations JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,quadratic-equation-and-inequalities JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,functions JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,other JEE Main 2025 (23 Jan Shift 1),Mathematics,22,"If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to",117.0,22,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,indefinite-integrals JEE Main 2025 (23 Jan Shift 1),Mathematics,23,"If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to",30.0,23,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,differential-equations JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,binomial-theorem JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,parabola JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,differentiation JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,other JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,application-of-derivatives JEE Main 2025 (23 Jan Shift 1),Mathematics,24,The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to,612.0,24,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,matrices-and-determinants JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,3d-geometry JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,area-under-the-curves JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,complex-numbers JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,permutations-and-combinations JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,hyperbola JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,vector-algebra JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 1),Mathematics,25,"If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to",77.0,25,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,1,"The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is: (1) \( \sqrt{17} \) (2) \( \sqrt{15} \) (3) \( \sqrt{14} \) (4) \( \sqrt{13} \)",3.0,1,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,other JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,2,"Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is: (1) 15 (2) 24 (3) 18 (4) 12",4.0,2,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,other JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,ellipse JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,parabola JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,3,"Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below: (1) Both Statement I and Statement II are false (2) Statement I is true but Statement II is false (3) Both Statement I and Statement II are true (4) Statement I is false but Statement II is true",2.0,3,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,4,"Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals: (1) 48 (2) 55 (3) 62 (4) 47",2.0,4,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,properties-of-triangle JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,statistics JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,ellipse JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,5,"A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to: (1) 22 (2) 21 (3) 23 (4) 24",3.0,5,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,properties-of-triangle JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,6,"If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to: (1) 21 (2) 9 (3) 14 (4) 6",2.0,6,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,parabola JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,7,"\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: (1) \( \frac{2}{3} \sqrt{e} \) (2) \( \frac{3e}{5} \) (3) \( \frac{2e}{3} \) (4) \( \frac{3e}{5} \)",3.0,7,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,8,"Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is: (1) \( \sqrt{7} \) (2) 14 (3) 3 (4) 7",4.0,8,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,ellipse JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,9,"The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is: (1) \( \frac{5}{3} \sqrt{15} \) (2) \( \frac{1}{3} \sqrt{15} \) (3) \( \frac{2}{3} \sqrt{15} \) (4) \( \sqrt{15} \)",3.0,9,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,statistics JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,10,"The system of equations \( x + 2y + 5z = 9 \), has no solution if: (x) \( x + 5y + \lambda z = \mu \), (1) \( \lambda = 15, \mu \neq 17 \) (2) \( \lambda \neq 17, \mu = 18 \) (3) \( \lambda = 17, \mu \neq 18 \) (4) \( \lambda = 17, \mu = 18 \)",3.0,10,ellipse JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,logarithm JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,11,"Let the range of the function \[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \] be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is: (1) 11 (2) 8 (3) 10 (4) 9",1.0,11,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,12,"Let \( x = x(y) \) be the solution of the differential equation \[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then \[ \cos(x(2)) \] is equal to: (1) \( 1 - 2(\log_2 2)^2 \) (2) \( 1 - 2(\log_2 2) \) (3) \( 2(\log_2 2)^2 - 1 \) (4) \( 2(\log_2 2)^2 - 1 \)",4.0,12,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,ellipse JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,13,"A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is: (1) 196π (2) 256π (3) 225π (4) 128π",2.0,13,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,14,"The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: (1) 4 (2) 8 (3) 10 (4) 6",2.0,14,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,15,"Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals: (1) -1 (2) 2 (3) 1 (4) 0",1.0,15,properties-of-triangle JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,16,"If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then \[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals: (1) \( \frac{x^2}{12} \) (2) \( \frac{x^2}{4} \) (3) \( \frac{x^2}{16} \) (4) \( \frac{x^2}{8} \)",3.0,16,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,probability JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,17,"A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is: (1) \(7/10\) (2) \(4/5\) (3) \(23/30\) (4) \(3/5\)",2.0,17,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,trigonometric-ratio-and-identites JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,18,"Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is: (1) \(x^2 + y^2 - 10x + 9 = 0\) (2) \(x^2 + y^2 - 6x + 5 = 0\) (3) \(x^2 + y^2 - 4x + 3 = 0\) (4) \(x^2 + y^2 - 8x + 7 = 0\)",2.0,18,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,parabola JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,19,"If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to: (1) 18 (2) 13 (3) 8 (4) 20",2.0,19,circle JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (23 Jan Shift 2),Mathematics,20,"If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is: (1) 8 (2) 7 (3) 5 (4) 6",3.0,20,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,definite-integration JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,statistics JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,21,"The variance of the numbers 8, 21, 34, 47, \ldots, 320 is",3.0,21,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,sequences-and-series JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,sets-and-relations JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,quadratic-equation-and-inequalities JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,functions JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,other JEE Main 2025 (23 Jan Shift 2),Mathematics,22,"The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to",474.0,22,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,indefinite-integrals JEE Main 2025 (23 Jan Shift 2),Mathematics,23,"The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is",17280.0,23,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,differential-equations JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,binomial-theorem JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,parabola JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,differentiation JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,other JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,application-of-derivatives JEE Main 2025 (23 Jan Shift 2),Mathematics,24,"The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to",15.0,24,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,matrices-and-determinants JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,3d-geometry JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,area-under-the-curves JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,complex-numbers JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,permutations-and-combinations JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,hyperbola JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,vector-algebra JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,limits-continuity-and-differentiability JEE Main 2025 (23 Jan Shift 2),Mathematics,25,"Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to",31.0,25,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,1,"Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) $4$ (3) $4 - \sqrt{3}$ (4) $3$",2.0,1,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,other JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,2,"Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) $17$ (2) $21$ (3) $56$ (4) $42$",2.0,2,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,other JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,ellipse JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,parabola JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,3,"Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$",2.0,3,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,4,"If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) $57$ (2) $59$ (3) $55$ (4) $56$",1.0,4,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,properties-of-triangle JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,statistics JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,ellipse JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,5,"For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is: (1) $20$ (2) $10$ (3) $35$ (4) $70$",3.0,5,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,properties-of-triangle JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,6,"The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to (1) $14$ (2) $21$ (3) $28$ (4) $7$",1.0,6,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,parabola JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,7,"Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is (1) $5$ (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) $10$",2.0,7,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,8,"Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) $84$ (2) $113$ (3) $91$ (4) $101$",3.0,8,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,ellipse JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,9,"If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to",1.0,9,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,statistics JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,10,"For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4",3.0,10,ellipse JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,logarithm JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,11,"The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3",4.0,11,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,12,"Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25",4.0,12,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,ellipse JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,13,"Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to (1) 5 (2) 3 (3) 4 (4) 6",3.0,13,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,14,"If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is (1) \(I(19, 27)\) (2) \(I(9, 1)\) (3) \(I(1, 13)\) (4) \(I(9, 13)\)",4.0,14,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,15,"\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is (1) \(\frac{9}{17}\) (2) \(\frac{9}{17}\) (3) \(\frac{9}{17}\) (4) \(\frac{8}{17}\)",2.0,15,properties-of-triangle JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,16,"Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to (1) 92 (2) 118 (3) 102 (4) 108",2.0,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,probability JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,17,"Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to (1) \(\sqrt{15} \div 2\) (2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\) (3) \(2\sqrt{2}\) (4) \(\sqrt{\frac{14}{3}}\)",2.0,17,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,trigonometric-ratio-and-identites JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,18,\(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is:,4.0,18,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,parabola JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,19,"Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is: (1) \( \frac{90}{11} \) (2) \( \frac{85}{11} \) (3) \( \frac{61}{12} \) (4) \( \frac{567}{121} \)",4.0,19,circle JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 1),Mathematics,20,"Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{C} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to (1) \( \sqrt{\frac{11}{6}} \) (2) \( \frac{1}{3\sqrt{2}} \) (3) \( 16 \) (4) \( 18 \)",1.0,20,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,definite-integration JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,statistics JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,21,"Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______.",5120.0,21,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,sequences-and-series JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,sets-and-relations JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,quadratic-equation-and-inequalities JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,functions JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,other JEE Main 2025 (24 Jan Shift 1),Mathematics,22,"If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______.",14.0,22,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,indefinite-integrals JEE Main 2025 (24 Jan Shift 1),Mathematics,23,"Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If \[ A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______.",44.0,23,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,differential-equations JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,binomial-theorem JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,parabola JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,differentiation JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,other JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,application-of-derivatives JEE Main 2025 (24 Jan Shift 1),Mathematics,24,"Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______.",19.0,24,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,matrices-and-determinants JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,3d-geometry JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,area-under-the-curves JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,complex-numbers JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,permutations-and-combinations JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,hyperbola JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,vector-algebra JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 1),Mathematics,25,"The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______.",125.0,25,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,1,"Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to: (1) 8750 (2) 9100 (3) 8925 (4) 8575",3.0,1,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,other JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,2,"If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to: $14x + 3y + \mu z = 33$ (1) 13 (2) 10 (3) 12 (4) 11",3.0,2,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,other JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,ellipse JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,parabola JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,3,"Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to: (1) 4 (2) 8 (3) 6 (4) 2",2.0,3,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,4,"The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: (1) $1 - \log_e 2$ (2) $\log_e 2$ (3) $1 + \log_e 2$ (4) $2 \log_e 2 - 1$",1.0,4,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,properties-of-triangle JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,statistics JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,ellipse JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,5,"The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is: $25x + 101y = 176$ (1) $48x + 25y = 169$ (2) $5x + 16y = 31$ (3) $4x + 122y = 134$ (4) $4x + 122y = 134$",,5,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,properties-of-triangle JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,6,"Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to: (1) 44 (2) 22 (3) 33 (4) 55",,6,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,parabola JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,7,"Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to: (1) 39 (2) 19 (3) 29 (4) 23",,7,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,8,"If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is: (1) $\frac{6}{7}$ (2) 6 (3) $\frac{1}{7}$ (4) 1",,8,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,ellipse JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,9,"Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to: (1) 6 (2) 8 (3) 9 (4) 7",2.0,9,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,statistics JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,10,Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:,3.0,10,ellipse JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,logarithm JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,11,"Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to: (1) 3 (2) 4 (3) 1 (4) 6",1.0,11,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,12,"Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is: (1) \(2\sqrt{14}\) (2) \(\mathbf{v}\) (3) \(\sqrt{7}\) (4) \(2\sqrt{7}\)",1.0,12,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,ellipse JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,13,"The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: (1) 1 (2) 0 (3) 2 (4) 3",,13,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,14,"The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto",,14,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,15,"In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: (1) 525 (2) 510 (3) 515 (4) 505",,15,properties-of-triangle JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,16,"Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to: (1) 22 (2) 20 (3) 21 (4) 19",,16,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,probability JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,17,"Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to: (1) 420 (2) 360 (3) 160 (4) 280",2.0,17,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,trigonometric-ratio-and-identites JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,18,"For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to: (1) 16 (2) 25 (3) 9 (4) 36",1.0,18,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,parabola JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,19,"If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\)",2.0,19,circle JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (24 Jan Shift 2),Mathematics,20,"If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to: (1) $\pi$ (2) 0 (3) $\pi - (\alpha + \beta + \gamma)$ (4) $3\pi$",1.0,20,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,definite-integration JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,statistics JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,21,"Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________.",,21,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,sequences-and-series JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,sets-and-relations JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,quadratic-equation-and-inequalities JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,functions JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,other JEE Main 2025 (24 Jan Shift 2),Mathematics,22,"If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________.",,22,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,indefinite-integrals JEE Main 2025 (24 Jan Shift 2),Mathematics,23,"Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________.",,23,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,differential-equations JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,binomial-theorem JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,parabola JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,differentiation JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,other JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,application-of-derivatives JEE Main 2025 (24 Jan Shift 2),Mathematics,24,"Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________.",,24,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,matrices-and-determinants JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,3d-geometry JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,area-under-the-curves JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,complex-numbers JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,permutations-and-combinations JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,hyperbola JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,vector-algebra JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,limits-continuity-and-differentiability JEE Main 2025 (24 Jan Shift 2),Mathematics,25,"Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.",55.0,25,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,1,"Let \( O \) be the origin, the point \( A \) be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point \( B(z_2) \) be such that \( \sqrt{3} |z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then (1) area of triangle ABO is \( \frac{11}{3} \) (2) ABO is an obtuse angled isosceles triangle (3) area of triangle ABO is \( \frac{11}{4} \) (4) ABO is a scalene triangle",2.0,1,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,other JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,2,"Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is (1) 545 (2) 715 (3) 735 (4) 675",4.0,2,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,other JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,ellipse JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,parabola JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,3,"Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is (1) \( \frac{73}{8} \) (2) \( \frac{25}{9} \) (3) \( \frac{16}{8} \) (4) \( \frac{75}{8} \)",1.0,3,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,4,"The sum of all local minimum values of the function \[ f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\ \frac{1}{12}(x - 4)(x - 5), & x > 2 \end{cases} \] is (1) \( \frac{137}{72} \) (2) \( \frac{131}{72} \) (3) \( \frac{137}{72} \) (4) \( \frac{167}{72} \)",1.0,4,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,properties-of-triangle JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,statistics JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,ellipse JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,5,"Let \( ^nC_{r-1} = 28, ^nC_r = 56 \) and \( ^nC_{r+1} = 70 \). Let \( A(4 \cos t, 4 \sin t), B(2 \sin t, -2 \cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals (1) 6 (2) 18 (3) 8 (4) 20",4.0,5,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,properties-of-triangle JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,6,"Let the equation of the circle, which touches \( x \)-axis at the point \( (a, 0) \), \( a > 0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^2 + y^2 - \alpha x + \beta y + \gamma = 0 \). If the circle lies below \( x \)-axis, then the ordered pair \((2a, b^2)\) is equal to (1) \( (\gamma, \beta^2 - 4\alpha) \) (2) \( (\alpha, \beta^2 + 4\gamma) \) (3) \( (\gamma, \beta^2 + 4\alpha) \) (4) \( (\alpha, \beta^2 - 4\gamma) \)",4.0,6,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,parabola JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,7,"If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to (1) \( 1.81\sqrt{2} \) (2) \( 41 \) (3) \( 82 \) (4) \( \frac{81}{2} \)",4.0,7,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,8,"Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals",2.0,8,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,ellipse JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,9,"If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to \[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8. \]",1.0,9,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,statistics JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,10,"\(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to: \[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}. \]",2.0,10,ellipse JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,logarithm JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,11,"Let \(A(x, y, z)\) be a point in \(xy\)-plane, which is equidistant from three points \((0, 3, 2), (2, 0, 3)\) and \((0, 0, 1)\). Let \(B = (1, 4, -1)\) and \(C = (2, 0, -2)\). Then among the statements (S1) : \(\triangle ABC\) is an isosceles right angled triangle, and (S2) : the area of \(\triangle ABC\) is \(\frac{9\sqrt{2}}{2}\), \[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}. \]",3.0,11,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,12,"The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is \[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}. \]",2.0,12,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,ellipse JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,13,"The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is \[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2}) \]",3.0,13,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,14,"Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then \(5m \sum_{r=m}^{m+2} T_r\) is equal to \[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112. \]",2.0,14,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,15,"Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \(x\) denote the number of defective oranges, then the variance of \(x\) is \[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}. \]",1.0,15,properties-of-triangle JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,16,"Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to \[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2. \]",1.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,probability JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,17,"If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals \[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100. \]",4.0,17,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,trigonometric-ratio-and-identites JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,18,"Let \(\{a_n\}\) be a sequence such that \(a_0 = 0, a_1 = \frac{1}{2}\) and \(2a_{n+2} = 5a_{n+1} - 3a_n, n = 0, 1, 2, 3, \ldots\). Then \(\sum_{k=1}^{100} a_k\) is equal to",2.0,18,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,parabola JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,19,"The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is (1) 4608 (2) 5720 (3) 5719 (4) 4607",4.0,19,circle JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 1),Mathematics,20,"The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric",2.0,20,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,definite-integration JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,statistics JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,21,"Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals \( \frac{5}{2} \)",5.0,21,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,sequences-and-series JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,sets-and-relations JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,quadratic-equation-and-inequalities JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,functions JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,other JEE Main 2025 (28 Jan Shift 1),Mathematics,22,"Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S = \{ -3, -2, -1, 1, 2 \} \). Let \[ S_1 = \{ A = [a_{ij}] \in M : A = A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_2 = \{ A = [a_{ij}] \in M : A = -A^T \text{ and } a_{ij} \in S, \forall i, j \}, \\ S_3 = \{ A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 \text{ and } a_{ij} \in S, \forall i, j \}. \] If \( n(S_1 \cup S_2 \cup S_3) = 125\alpha \), then \( \alpha \) equals \( \frac{5}{2} \)",1613.0,22,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,indefinite-integrals JEE Main 2025 (28 Jan Shift 1),Mathematics,23,"If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is \( \frac{5}{2} \)",5.0,23,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,differential-equations JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,binomial-theorem JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,parabola JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,differentiation JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,other JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,application-of-derivatives JEE Main 2025 (28 Jan Shift 1),Mathematics,24,"Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1}(i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to \( \frac{5}{\pi} \)",54.0,24,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,matrices-and-determinants JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,3d-geometry JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,area-under-the-curves JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,complex-numbers JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,permutations-and-combinations JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,hyperbola JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,vector-algebra JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 1),Mathematics,25,"Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}|, |\vec{c} - 2\vec{a}|^2 = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2 \) is equal to \( \frac{5}{2} \)",6.0,25,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,1,"Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is: (1) 127 (2) 258 (3) 65 (4) 2049",3.0,1,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,other JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,2,"If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: (1) 26 (2) 18 (3) 23 (4) 16",1.0,2,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,other JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,ellipse JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,parabola JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,3,"Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is: (1) 2 (2) 9/2 (3) 1 (4) 0",3.0,3,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,4,"Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to: (1) 283 (2) 287 (3) 295 (4) 299",1.0,4,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,properties-of-triangle JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,statistics JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,ellipse JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,5,"Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: (1) \( (-\infty, -1] \cup [0, \infty) \) (2) \( (-\infty, -1] \cup [1, \infty) \) (3) \( (-\infty, \infty) \) (4) \( (-\infty, \infty) \) \( \setminus \{0\} \)",3.0,5,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,properties-of-triangle JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,6,"Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is: (1) \( \frac{1}{2} \) (2) \( \frac{1}{4} \) (3) \( \frac{3}{5} \) (4) \( \frac{1}{5} \)",1.0,6,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,parabola JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,7,"If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to: (1) 10 (2) 4 (3) 2 (4) 8",4.0,7,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,8,"Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is: (1) 270 (2) 340 (3) 320 (4) 310",4.0,8,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,ellipse JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,9,"Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:",2.0,9,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,statistics JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,10,"Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is: (1) $\frac{4}{15}$ (2) $\frac{1}{3}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$",1.0,10,ellipse JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,logarithm JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,11,"Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to: (1) 11 (2) 13 (3) 15 (4) 9",1.0,11,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,12,"For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is: (1) 540 (2) 675 (3) 1350 (4) 135",2.0,12,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,ellipse JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,13,"Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is: (1) 7 (2) 6 (3) 1 (4) 9",2.0,13,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,14,"If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line: (1) $x + 9y = 36$ (2) $4x - 9y = 12$ (3) $6x - 9y = 20$ (4) $9x - 9y = 32$",3.0,14,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,15,"If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to: (1) $4 \log_e 2 - 2$ (2) $2 - \log_e x$ (3) $\log_e 2 + 2$ (4) $4 \log_e 2 + 2$",1.0,15,properties-of-triangle JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,16,"The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: (1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$ (2) $\frac{\pi}{2} - \frac{1}{3}$ (3) $\frac{\pi}{2} - \frac{1}{3}$ (4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$",2.0,16,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,probability JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,17,"The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is: (1) 54 (2) 44 (3) 41 (4) 66",4.0,17,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,trigonometric-ratio-and-identites JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,18,"If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is: (1) 20 (2) 22 (3) 18 (4) 26",2.0,18,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,parabola JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,19,"If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to: (1) $-2$ (2) $6$ (3) $-6$ (4) $2$",4.0,19,circle JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (28 Jan Shift 2),Mathematics,20,"Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is: (1) $-2\sqrt{10}$ (2) $12$ (3) $6$ (4) $-6$",3.0,20,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,definite-integration JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,statistics JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,21,"Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is",14.0,21,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,sequences-and-series JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,sets-and-relations JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,quadratic-equation-and-inequalities JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,functions JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,other JEE Main 2025 (28 Jan Shift 2),Mathematics,22,"The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is",64.0,22,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,indefinite-integrals JEE Main 2025 (28 Jan Shift 2),Mathematics,23,"If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to",4.0,23,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,differential-equations JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,binomial-theorem JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,parabola JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,differentiation JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,other JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,application-of-derivatives JEE Main 2025 (28 Jan Shift 2),Mathematics,24,"The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to",20.0,24,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,matrices-and-determinants JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,3d-geometry JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,area-under-the-curves JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,complex-numbers JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,permutations-and-combinations JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,hyperbola JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,vector-algebra JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,limits-continuity-and-differentiability JEE Main 2025 (28 Jan Shift 2),Mathematics,25,Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to,1.0,25,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,1,"Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to: (1) 100 (2) 120 (3) 110 (4) 90",1.0,1,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,other JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,2,"Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is: (1) 90 (2) 84 (3) 122 (4) 108",1.0,2,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,other JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,ellipse JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,parabola JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,3,"The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is: (1) 2 (2) 3 (3) 1 (4) 4",4.0,3,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,4,"Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is: (1) both reflexive and transitive but not symmetric (2) an equivalence relation (3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive",2.0,4,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,properties-of-triangle JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,statistics JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,ellipse JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,5,"Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to: (1) 392 (2) 384 (3) 192 (4) 96",3.0,5,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,properties-of-triangle JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,6,"Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is: (1) 173 (2) 164 (3) 158 (4) 161",4.0,6,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,parabola JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,7,"Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point: (1) (5, 17, 4) (2) (2, 8, 5) (3) (8, 26, 12) (4) (-1, -1, 1)",3.0,7,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,8,"Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is: (1) 462 (2) 77 (3) 154 (4) 308",4.0,8,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,ellipse JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,9,"The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: (1) 3 \log_e 4 (2) 4 \log_e 3 (3) 6 \log_e 4 (4) 2 \log_e 3",2.0,9,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,statistics JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,10,"Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals: \[ \begin{align*} (1) & \quad 4\sqrt{6} \\ (2) & \quad 6\sqrt{6} \\ (3) & \quad 18\sqrt{6}/5 \\ (4) & \quad 24\sqrt{6}/5 \\ \end{align*} \]",4.0,10,ellipse JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,logarithm JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,11,"Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to: \[ \begin{align*} (1) & \quad 288 \\ (2) & \quad 222 \\ (3) & \quad 242 \\ (4) & \quad 262 \\ \end{align*} \]",3.0,11,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,12,"Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is: \[ \begin{align*} (1) & \quad 13 \\ (2) & \quad 10 \\ (3) & \quad 3 \\ (4) & \quad 7 \\ \end{align*} \]",4.0,12,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,ellipse JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,13,"Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) be two lines. Let \( L_3 \) be a line passing through the point \((\alpha, \beta, \gamma)\) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals: \[ \begin{align*} (1) & \quad 20 \\ (2) & \quad 18 \\ (3) & \quad 25 \\ (4) & \quad 16 \\ \end{align*} \]",3.0,13,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,14,"Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to: \[ \begin{align*} (1) & \quad 1280 \\ (2) & \quad 1295 \\ (3) & \quad 1215 \\ (4) & \quad 1040 \\ \end{align*} \]",1.0,14,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,15,"Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to: \[ \begin{align*} (1) & \quad 47 \\ (2) & \quad 36 \\ (3) & \quad 47 \\ (4) & \quad 40 \\ \end{align*} \]",2.0,15,properties-of-triangle JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,16,"The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is: \[ \begin{align*} (1) & \quad 4/3 \\ (2) & \quad 2 \\ (3) & \quad 7/3 \\ (4) & \quad 5/3 \\ \end{align*} \]",4.0,16,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,probability JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,17,"The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is: \[ \begin{align*} (1) & \quad 2184 \\ (2) & \quad 2196 \\ (3) & \quad 2148 \\ (4) & \quad 2172 \\ \end{align*} \]",1.0,17,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,trigonometric-ratio-and-identites JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,18,"Let \( y = y(x) \) be the solution of the differential equation \[ \cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\] If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to: \[ \begin{align*} (1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\ (2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\ (3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\ (4) & \frac{1}{\log_2 (4)} \end{align*} \]",1.0,18,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,parabola JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,19,"Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to: \[ \begin{align*} (1) & \sqrt{14} \\ (2) & 3\sqrt{7} \\ (3) & 2\sqrt{14} \\ (4) & 5\sqrt{7} \end{align*} \]",3.0,19,circle JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,straight-lines-and-pair-of-straight-lines JEE Main 2025 (29 Jan Shift 1),Mathematics,20,"Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to: \[ \begin{align*} (1) & 16 \\ (2) & 12 \\ (3) & 14 \\ (4) & 18 \end{align*} \]",3.0,20,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,definite-integration JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,statistics JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,21,Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______.,5.0,21,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,sequences-and-series JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,sets-and-relations JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,quadratic-equation-and-inequalities JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,functions JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,other JEE Main 2025 (29 Jan Shift 1),Mathematics,22,"Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______.",112.0,22,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,indefinite-integrals JEE Main 2025 (29 Jan Shift 1),Mathematics,23,"The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.",1405.0,23,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,differential-equations JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,binomial-theorem JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,parabola JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,differentiation JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,other JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,application-of-derivatives JEE Main 2025 (29 Jan Shift 1),Mathematics,24,"Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______.",2.0,24,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,matrices-and-determinants JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,3d-geometry JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,area-under-the-curves JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,complex-numbers JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,permutations-and-combinations JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,hyperbola JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,vector-algebra JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,limits-continuity-and-differentiability JEE Main 2025 (29 Jan Shift 1),Mathematics,25,"Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which \[ \lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1 \] is equal to ______.",24.0,25,limits-continuity-and-differentiability