JEE Main 2025 (24 Jan Shift 2)

63 Questions • 75 Minutes • 2025

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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 :

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If the system of equations $x+2y-3z=2$, $2x+\lambda y+5z=5$, $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to :

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Let $A=\left\{x\in(0,\pi)-\left\{\frac\pi2\right\}:\log_{(2/\pi)}|\sin x|+\log_{(2/\pi)}|\cos x|=2\right\}$ and $B=\{x\ge0:\sqrt x(\sqrt x-4)-3|\sqrt x-2|+6=0\}$. Then $n(A\cup B)$ is equal to :

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The area of the region enclosed by the curves $y=e^x$, $y=|e^x-1|$ and $y$-axis is:

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The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid-point is $(3,1)$ is :

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Let the points $\left(\frac{11}2,\alpha\right)$ 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 :

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Let $f:(0,\infty)\to\mathbf R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2f'(x)=2xf(x)+3$, with $f(1)=4$. Then $2f(2)$ is equal to :

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If $7=5+\frac17(5+\alpha)+\frac1{7^2}(5+2\alpha)+\frac1{7^3}(5+3\alpha)+\dots\infty$, then the value of $\alpha$ is :

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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 :

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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 :

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Let the position vectors of three vertices of a triangle be $4\vec p+\vec q-3\vec r$, $-5\vec p+\vec q+2\vec r$ and $2\vec p-\vec q+2\vec r$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec p+\vec q+\vec r}4$ and $\alpha\vec p+\beta\vec q+\gamma\vec r$ respectively, then $\alpha+2\beta+5\gamma$ is equal to :

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Let $\vec a=3\hat i-\hat j+2\hat k$, $\vec b=\vec a\times(\hat i-2\hat k)$ and $\vec c=\vec b\times\hat k$. Then the projection of $\vec c-2\hat j$ on $\vec a$ is :

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The number of real solution(s) of the equation $x^2+3x+2=\min\{|x-3|,|x+2|\}$ is :

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The function $f:(-\infty,\infty)\to(-\infty,1)$, defined by $f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}$ is :

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In an arithmetic progression, if $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ is equal to :

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Suppose A and B are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2A=5B$, then $n$ is equal to :

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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 :

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For some $a,b$, let $f(x)=\begin{vmatrix}a+\frac{\sin x}x&1&b\\a&1+\frac{\sin x}x&b\\a&1&b+\frac{\sin x}x\end{vmatrix}$, $x\neq0$, $\lim_{x\to0}f(x)=\lambda+\mu a+\nu b$. Then $(\lambda+\mu+\nu)^2$ is equal to :

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If the equation of the parabola with vertex $V\left(\frac32,3\right)$ 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 :

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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 :

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Paper Info

Exam: JEE
Year: 2025
Questions: 63
Duration: 75 mins

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