JEE MAIN 2025 Previous Year Questions (PYQs) – Page 3 of 38

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 $(\overrightarrow{OQ}\cdot\overrightarrow{OA})-\dfrac{1}{5}\lvert\overrightarrow{OP}\times\overrightarrow{OA}\rvert^{2}=10$, then the value of $r$ is:

If the sum of the first $20$ terms of the series $\dfrac{4\cdot1}{4+3\cdot1^{2}+1^{4}}+\dfrac{4\cdot2}{4+3\cdot2^{2}+2^{4}}+\dfrac{4\cdot3}{4+3\cdot3^{2}+3^{4}}+\dfrac{4\cdot4}{4+3\cdot4^{2}+4^{4}}+\cdots$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to:

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 $\frac{m}{n}$, where $m$, $n$ are coprime numbers, then $m+n$ is equal to :

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\ \dfrac{x^{2}}{4^{2}}+\dfrac{y^{2}}{3^{2}}=1$ at the points $A$ and $B$. Let the line $y=x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $ABCD$ is

Let X = ℝ × ℝ. Define a relation R on X by (a₁,b₁) R (a₂,b₂) ⇔ b₁ = b₂. Statement I: R is an equivalence relation. Statement II: For some (a,b) ∈ X, the set S = { (x,y) ∈ X : (x,y) R (a,b) } represents a line parallel to y = x.

The centre of a circle $C$ is at the centre of the ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a>b$. Let $C$ pass through the foci $F_{1}$ and $F_{2}$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $PF_{1}F_{2}$ is $30$ and the length of the major axis of $E$ is $17$, then the distance between the foci of $E$ is

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

A line passing through the point $A(-2,0)$ touches the parabola $P: y^2=x-2$ at the point $B$ in the first quadrant. The area of the region bounded by the line $\overline{AB}$, parabola $P$ and the $x$-axis is:

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

Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\alpha+i\beta$, where $i=\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then $p+q$ is equal to: