JEE MAIN Mathematics Medium 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:

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 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 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 ${\sum }_{k=1}^n{a}_k=6n^3$, then ${\left(\frac{{\sum }_{k=1}^6(a_{k+1}-a_k)}{36}\right)}^2$ is equal to ______.

The number of elements in the set $S=\{(r,k):k\in \mathbb{Z}\text{ and }{}^{36}{C}_{r+1}=\frac{6\cdot {}^{35}{C}_r}{k^2-3}\}$ is:

Let $\alpha ,\beta$ be the roots of the equation $x^2-3x+r=0$, and $\frac{\alpha }{2},2\beta$ be the roots of the equation $x^2+3x+r=0$. If the roots of the equation $x^2+6x=m$ are $2\alpha +\beta +2r$ and $\alpha -2\beta -\frac{r}{2}$, then $m$ is equal to:

If the system of equations $x+5y+6z=4$, $2x+3y+4z=7$, $x+6y+az=b$ has infinitely many solutions, then the point $(a,b)$ lies on the line

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:

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:

Let $y=y(x)$ be the solution curve of the differential equation $(1+\sin x)\frac{dy}{dx}+(y+1)\cos x=0$, $y(0)=0$. If the curve passes through the point $(\alpha ,-\frac{1}{2})$, then a value of $\alpha$ is:

Let $a,b,c\in \{1,2,3,4\}$. If the probability that $a{x}^2+2\sqrt{2}bx+c>0$ for all $x\in \mathbb{R}$ is $\frac{m}{n}$ with $\gcd (m,n)=1$, then $m+n$ is equal to ______.

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:

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 $\alpha ,\beta \in \mathbb{R}$ be such that the system of linear equations $x+2y+z=5$, $2x+y+\alpha z=5$, $8x+4y+\beta z=18$ has no solution. Then $\frac{\beta }{\alpha }$ is equal to:

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:

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to: