List of top Questions asked in JEE Main

Let PQ be a focal chord of the parabola y² = 36x of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM:MQ = 3:1. Then which of the following points does $\underline{NOT}$ lie on the line passing through M and perpendicular to the line PQ?

Consider ellipse $ E_k : \frac{x^2}{k} + \frac{y^2}{k} = 1 $, for $ k = 1, 2, \dots, 20 $. Let $ C_k $ be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse $ E_k $. If $ r_k $ is the radius of the circle $ C_k $, then the value of $ \sum_{k=1}^{20} r_k^2 $ is:

If a point $P (\alpha, \beta, \gamma)$ satisfying $(\alpha\,\, \beta\,\, \gamma) \begin{pmatrix} 2 & 10 & 8 \\9 & 3 & 8 \\8 & 4 & 8\end{pmatrix}=(0\,\,0\,\,0) $ lies on the plane $2 x+4 y+3 z=5$, then $6 \alpha+9 \beta+7 \gamma$ is equal to :

The locus of the mid points of the chords of the circle $C_1:(x-4)^2+(y-5)^2=4$ which subtend an angle $\theta_i$ at the centre of the circle $C_1$, is a circle of radius $r_i$ If $\theta_1=\frac{\pi}{3}, \theta_3=\frac{2 \pi}{3}$ and $r_1^2=r_2^2+r_3^2$, then $\theta_2$ is equal to

Let $y=y(x)$ be the solution of the differential equation $x^3 d y+(x y-1) d x=0, x>0$, $y\left(\frac{1}{2}\right)=3- e$ Then $y(1)$ is equal to

The number of solutions of cos4θ – 2cos2θ + 3sin6θ + 1 = 0 in [0, 2π] is

Area bounded by larger part in I quadrant by x = 4y² , x = 2 and y = x is A then 3A equals

Let $x = 2$ be a root of the equation $x^2 + px + q = 0$ and $f(x)=\begin{cases}\frac{1-cos(x^2-4px+q^2+8q+16)}{(x-2p)^4} & x≠2p\\0 & x=2p\end{cases}$.
Then $lim_{x \rightarrow 2p}[f(x)]$, where [·] denotes greatest integer function, is

Let B and C be the two points on the line $y + x = 0$ such that B and C are symmetric with respect to the origin. Suppose A is a point on $y – 2x = 2$ such that $\Delta ABC$ is an equilateral triangle. Then, the area of the $\Delta ABC$ is

Let sets A and B have 5 elements each. Let mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is:

Two dice are thrown independently Let$A$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $B$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even Then :

The compound statement $(-(P \wedge Q)) \vee((-P) \wedge Q) \Rightarrow((-P) \wedge(-Q))$ is equivalent to

Let $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b} \cdot \vec{c}=0$. Consider the following two statements:

(A) $|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|$ for all $\lambda \in R$

(B) $\vec{a}$ and $\vec{c}$ are always parallel. Then. is

Plane P₃ is passing through (1,1,1) and line of intersection of P₁ and P₂ where $P_{1}: 2x - y + z = 5$ and $P_{2}: x + 3y + 2z + 2 = 0$. Then distance of (1,1,10) from P₃ is:

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of its squares of first three terms is 33033, then the sum of these three terms is equal to

Let $ f: [2, 4] \to \mathbb{R} $ be a differentiable function such that $ (x \log x) f'(x) + (\log x) f(x) \geq 1 $, $ x \in [2, 4] $ with $ f(2) = \frac{1}{2} $ and $ f(4) = \frac{1}{4} $. Consider the following two statements:
$ (A) \quad f(x) \geq 1 \quad \text{for all} \quad x \in [2, 4] $
$ (B) \quad f(x) \leq \frac{1}{8} \quad \text{for all} \quad x \in [2, 4] $ Then,

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$ If the curve passes through the point $(1,2)$ and $(8,1)$, then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to

If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>$, then

Let P = $\left[\begin{matrix} \frac{\sqrt3}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt3}{2} \end{matrix}\right]$ A = $\left[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right]$ and Q = PAP^T. If P^TQ²⁰⁰⁷P = $\left[\begin{matrix} a & b \\ c & d \end{matrix}\right]$, then 2a+b-3c-4d equal to

The slope of tangent at any point (x, y) on a curve y = y(x) is $\frac{x^2+y^2}{2xy},x>0$ If y(2) = 0, then a value of y(8) is

if $2x^y+3y^x=20$ then $\frac{dy}{dx}$ at (2,2) is equal to:

Let$ S={x∈R:0<x<1 and\ 2 tan−1\frac{(1+x)}{(1−x)}=cos^{−1}\frac{(1-x^2)}{(1+x^2)}}$. If n(S) denotes the number of elements in S then :

The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$, is

Let the plane containing the line of intersection of the planes $P 1: x+(\lambda+4) y+z=1$ and $P 2: 2 x+$ $y+z=2$ pass through the points $(0,1,0)$ and $(1,0,1)$ Then the distance of the point $(2 \lambda, \lambda,-\lambda)$ from the plane $P 2$ is