JEE Main 2025 (24 Jan Shift 1)

70 Questions • 75 Minutes • 2025

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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 $(1/6)^{\text{th}}$ of the perimeter of $C$, then $\beta-\sqrt3\alpha$ is equal to

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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 $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Then the area (in sq. units) of $\triangle ABC$ is:

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Let the product of the focal distances of the point $\left(\sqrt3,\frac12\right)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$, be $\frac74$. Then the absolute difference of the eccentricities of two such ellipses is

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If the system of equations $2x-y+z=4$, $5x+\lambda y+3z=12$, $100x-47y+\mu z=212$ has infinitely many solutions, then $\mu-2\lambda$ is equal to

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For some $n\neq10$, let the coefficients of the 5th, 6th and 7th 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:

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The product of all the rational roots of the equation $(x^2-9x+11)^2-(x-4)(x-5)=3$, is equal to

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Let the line passing through the points $(-1,2,1)$ and parallel to the line $\frac{x-1}2=\frac{y+1}3=\frac z4$ 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

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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

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If $\alpha$ and $\beta$ are the roots of the equation $2z^2-3z-2i=0$, where $i=\sqrt{-1}$, then $16\cdot\operatorname{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)\cdot\operatorname{Im}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$ is equal to

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For a statistical data $x_1,x_2,\dots,x_{10}$ 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

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The area of the region $\{(x,y):x^2+4x+2\le y\le|x+2|\}$ is equal to

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Let $S_n=\frac12+\frac16+\frac1{12}+\frac1{20}+\dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026\,S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is

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Let $f:R-\{0\}\to R$ be a function such that $f(x)-6f\left(\frac1x\right)=\frac{35}{3x}-\frac52$. If the $\lim_{x\to0}\left(\frac1{\alpha x}+f(x)\right)=\beta;\ \alpha,\beta\in R$, then $\alpha+2\beta$ is equal to

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If $I(m,n)=\int_0^1x^{m-1}(1-x)^{n-1}dx,\ m,n>0$, then $I(9,14)+I(10,13)$ is

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$A$ 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

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Let $f(x)=\frac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac1{15}\right)+f\left(\frac2{15}\right)+\dots+f\left(\frac{59}{15}\right)\right)$ is equal to

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Let $y=y(x)$ be the solution of the differential equation $\left(xy-5x^2\sqrt{1+x^2}\right)dx+\left(1+x^2\right)dy=0,\ y(0)=0$. Then $y(\sqrt3)$ is equal to

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$\lim_{x\to0}\operatorname{cosec}x\left(\sqrt{2\cos^2x+3\cos x}-\sqrt{\cos^2x+\sin x+4}\right)$ is:

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Consider the region $R=\left\{(x,y):x\le y\le9-\frac{11}3x^2,\ x\ge0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is:

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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

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Paper Info

Exam: JEE
Year: 2025
Questions: 70
Duration: 75 mins

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