JEE MAIN 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:

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:

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 ____.

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:

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 $O$ 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 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 $A=\left[\begin{array}{ccc}1& 21& \alpha \end{array}\right]$ 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 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 the point of intersection of the lines $\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7}$ and $\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7}$ lies on the $xy$-plane, then the value of $a+b$ 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:

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=2$ and $|\vec{b}|=3$, then the maximum value of $3\left|3\vec{a}+2\vec{b}\right|+4\left|3\vec{a}-2\vec{b}\right|$ is:

Let $\alpha$ and $\alpha +2$, with $\alpha \in \mathbb{Z}$, be the roots of the quadratic equation $x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots +(x+n-1)(x+n+1)=4n$ for some $n\in \mathbb{N}$. Then $n+\alpha$ is equal to:

Let a line $L$ passing through the point $(1,1,1)$ be perpendicular to both the vectors $2\hat{i}+2\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+2\hat{k}$. If $P(a,b,c)$ is the foot of perpendicular from the origin on the line $L$, then the value of $34(a+b+c)$ 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:

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: