JEE MAIN 2026 Mathematics Previous Year Questions

Let $A=\{2,3,4,5,6\}$. Let $R$ be a relation on the set $A\times A$ given by $(x,y)R(z,w)$ if and only if $x$ divides $z$ and $y=w$. Then the number of elements in $R$ is ______.

Let the parabola $y=x^2+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2+q^2$ is:

If $\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)=K$, then the value of $\sin \left(\frac{10K\pi }{3}\right)$ is:

Let an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with $a<b$, pass through the point $(4,3)$ and have eccentricity $\frac{\sqrt{5}}{3}$. Then the length of its latus rectum is:

If the domain of the function $f(x)=\sqrt{{\log }_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty ,a]\cup \{b\}\cup [c,d)\cup (e,\infty )$, then the value of $a+b+c+d+e$ is ____.

The number of seven-digit numbers that can be formed by using the digits $1,2,3,5,7$ such that each digit is used at least once, is:

Let $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy=12$ such that the mid-point of the line segment $PQ$ is $\left(\frac{1}{2},-\frac{1}{2}\right)$. Then the area of the triangle $OPQ$ equals:

Let the circles $C_1:|z|=r$ and $C_2:|z-3-4i|=5$, $z\in \mathbb{C}$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min |z_1-z_2|=2$, then $\max |z_1-z_2|$ is equal to:

Let $A=\left[\begin{array}{ccc}1& 21& \alpha \end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 3\beta & 2\end{array}\right]$. If $A^2-4A+I=O$ and $B^2-5B-6I=O$, then among the two statements:
(S1): ${\left[(B-A)(B+A)\right]}^T=\left[\begin{array}{ccc}13& 157& 10\end{array}\right]$ (S2): $\det (\mathrm{adj}(A+B))=-5$
which one is correct?

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x+(k-1)y+3=0$ and $2x+k^{2}y-4=0$. If the line $x-y+2=0$ intersects the circle at the points $A$ and $B$, then $(AB{)}^2$ is equal to:

Let the vectors $\vec{a}=-\hat{i}+\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+\hat{k}$. For some $\lambda ,\mu \in \mathbb{R}$, let $\vec{c}=\lambda \vec{a}+\mu \vec{b}$. If $\vec{c}\cdot (3\hat{i}-6\hat{j}+2\hat{k})=10$ and $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=-2$, then $|\vec{c}{|}^2$ is equal to:

The sum $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots$ up to $8$ terms, is:

If $\underset{x\to 2}{\lim }\frac{\sin \left(x^3-5x^2+ax+b\right)}{\left(\sqrt{x-1}-1\right){\log }_e(x-1)}=m$, then $a+b+m$ is equal to:

For the grouped data with classes 5–10, 10–15, 15–20, 20–25, 25–30, 30–35 and corresponding frequencies $2,k,28,54,k+1,5$, if the mean of the data is $21$, then $k$ is one of the roots of the equation:

If the curve $y=f(x)$ passes through the point $(1,e)$ and satisfies the differential equation $dy=y(2+{\log }_{e}x)dx$, $x>0$, then $f(e)$ is equal to:

The number of critical points of the function $f(x)=\{\begin{array}{llc}\frac{|\sin x|}{x},& x\ne 01,& x=0\end{array}$ in the interval $(-2\pi ,2\pi )$ is equal to:

Let the midpoints of the sides of a triangle $ABC$ be $\left(\frac{5}{2},7\right)$, $\left(\frac{5}{2},3\right)$ and $(4,5)$. If its incenter is $(h,k)$, then $3h+k$ is equal to:

Let $[\cdot ]$ denote the greatest integer function. Then the value of ${\int }_0^3\frac{e^x+e^{-x}}{[x]!}dx$ is:

Let $x$ and $y$ be real numbers such that $50\left(\frac{2x}{1+3i}-\frac{y}{1-2i}\right)=31+17i$, where $i=\sqrt{-1}$. Then the value of $10(x-3y)$ is:

Let $A$ be the set of first $101$ terms of an A.P. whose first term is $1$ and common difference is $5$, and let $B$ be the set of first $71$ terms of an A.P. whose first term is $9$ and common difference is $7$. Then the number of elements in $A\cap B$ which are divisible by $3$ is: