JEE Main 2025 (23 Jan Shift 2)

68 Questions • 75 Minutes • 2025

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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 $\frac x1=\frac{y-2}2=\frac{z+3}3$ is :

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Let $A=\{(x,y)\in\mathbf R\times\mathbf R:|x+y|\ge3\}$ and $B=\{(x,y)\in\mathbf R\times\mathbf R:|x|+|y|\le3\}$. If $C=\{(x,y)\in A\cap B:x=0\text{ or }y=0\}$, then $\sum_{(x,y)\in C}|x+y|$ is :

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Let $X=\mathbf R\times\mathbf R$. Define a relation $R$ on $X$ as : $(a_1,b_1)R(a_2,b_2)\Leftrightarrow b_1=b_2$. 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 :

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Let $\int x^3\sin x\,dx=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac\pi2\right)+g'\left(\frac\pi2\right)\right)=\alpha\pi^3+\beta\pi^2+\gamma$, $\alpha,\beta,\gamma\in Z$, then $\alpha+\beta-\gamma$ equals :

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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 :

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If the square of the shortest distance between the lines $\frac{x-2}1=\frac{y-1}2=\frac{z+3}{-3}$ and $\frac{x+1}2=\frac{y+3}4=\frac{z+5}{-5}$ is $\frac mn$, where $m,n$ are coprime numbers, then $m+n$ is equal to :

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$\lim_{x\to\infty}\frac{\left(2x^2-3x+5\right)(3x-1)^{x/2}}{\left(3x^2+5x+4\right)\sqrt{(3x+2)^x}}$ is equal to :

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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 $(\overrightarrow{OQ}\cdot\overrightarrow{OA})-\frac15\left|\overrightarrow{OP}\times\overrightarrow{OA}\right|^2=10$, then the value of $r$ is :

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The length of the chord of the ellipse $\frac{x^2}4+\frac{y^2}2=1$, whose mid-point is $\left(1,\frac12\right)$, is :

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The system of equations $x+y+z=6$, $x+2y+5z=9$, $x+5y+\lambda z=\mu$, has no solution if

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Let the range of the function $f(x)=6+16\cos x\cdot\cos\left(\frac\pi3-x\right)\cdot\cos\left(\frac\pi3+x\right)\cdot\sin3x\cdot\cos6x$, $x\in\mathbf R$ be $[\alpha,\beta]$. Then the distance of the point $(\alpha,\beta)$ from the line $3x+4y+12=0$ is :

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Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{dx}{dy}\right)\sin\left(\frac xy\right)$, $y>0$ and $x(1)=\frac\pi2$. Then $\cos(x(2))$ is equal to :

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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 $81\ \mathrm{cm^3/min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac1{4\pi}$ cm/min. The surface area (in $\mathrm{cm^2}$) of the chocolate ball (without the ice-cream layer) is :

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The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac z{\bar z}+\frac{\bar z}z\right|=1$, is :

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Let $A=[a_{ij}]$ be $3\times3$ matrix such that $A\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix}$, $A\begin{bmatrix}4\\1\\3\end{bmatrix}=\begin{bmatrix}0\\1\\0\end{bmatrix}$ and $A\begin{bmatrix}2\\1\\2\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$, then $a_{23}$ equals :

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If $I=\int_0^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}dx$, then $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4x+\cos^4x}dx$ equals :

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A board has 16 squares (a $4\times4$ grid). Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

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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 :

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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 :

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If the area of the region $\{(x,y):-1\le x\le1,\ 0\le y\le a+e^{|x|}-e^{-x},\ a>0\}$ is $\frac{e^2+8e+1}e$, then the value of $a$ is :

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Paper Info

Exam: JEE
Year: 2025
Questions: 68
Duration: 75 mins

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