List of top Questions asked in JEE Main- 2021

Two small spheres each of mass 10 mg are suspended from a point by threads 0.5 m long. They are equally charged and repel each other to a distance of 0.20 m. The charge on each of the sphere is $\frac{a}{21} \times 10^{-8} C$.
The value of 'a' will be ______. [Given g = 10 ms–2]

Given that dy/dx = ye^x such that x = 0, y = e. The value of y(y > 0) when x = 1 will be

the angle of intersection of the curves y=x² and x=y² at (1,1) is

The sum of solutions of the equation cos$\frac{cos\,x}{1+sin\,x}$=|tan 2x|,x$\in$($-\frac{\pi}{2},\frac{\pi}{2}$)-($-\frac{\pi}{4},\frac{\pi}{4}$) is:

If $\sum_{r=1}^{50} \tan^{-1}\frac{1}{2r^2} = p$, then the value of $\tan p$ is :

The integral ∫ (2x - 1) cos√((2x - 1)² + 5) / √(4x² - 4x + 6) dx is equal to : (where c is a constant of integration)

Let $\frac{\sin A}{\sin B} = \frac{\sin(A - C)}{\sin(C - B)}$, where A, B, C are angles of a triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A $\times$ B. Then :

If $ \alpha + \beta + \gamma = 2\pi $, then the system of equations
$ x + (\cos \gamma)y + (\cos \beta)z = 0 $
$ (\cos \gamma)x + y + (\cos \alpha)z = 0 $
$ (\cos \beta)x + (\cos \alpha)y + z = 0 $
has :

The distance of the point $(1, -2, 3)$ from the plane $x - y + z = 5$ measured parallel to a line, whose direction ratios are $2, 3, -6$ is :

Let α be the angle between the lines whose direction cosines satisfy the equations l + m - n = 0 and l² + m² - n² = 0. Then the value of sin⁴α + cos⁴α is :

A possible value of 'x', for which the ninth term in the expansion of
$\left\{ 3^{\log_3\sqrt{25^{x-1}+7}} + 3^{(-\frac{1}{8})\log_3(5^{x-1}+1)} \right\}^{10}$ in the increasing powers of $3^{(-\frac{1}{8})\log_3(5^{x-1}+1)}$ is equal to 180, is :

Let a tangent be drawn to the ellipse x²/27 + y² = 1 at (3$\sqrt{3}$ cosθ, sinθ) where θ ∈ (0, π/2). Then the value of θ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

A circle C touches the line $x=2y$ at the point (2, 1) and intersects the circle C$_1$ : $x^2 + y^2 + 2y - 5 = 0$ at two points P and Q such that PQ is a diameter of C$_1$. Then the diameter of C is :

If $x^2 + 9y^2 - 4x + 3 = 0, x, y \in \mathbb{R}$, then $x$ and $y$ respectively lie in the intervals :

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = $\sqrt{5}$ inches and $\angle PCB = \tan^{-1}(2)$. The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is :

Let $ f : \mathbb{R} \to \mathbb{R} $ be a function defined as \[ f(x) = \begin{cases} \dfrac{\sin\!\big((a+1)x\big) + \sin 2x}{2x}, & \text{if } x < 0, \\[6pt] b, & \text{if } x = 0, \\[6pt] \dfrac{\sqrt{x + b x^3} - \sqrt{x}}{b\, x^{5/2}}, & \text{if } x > 0. \end{cases} \] If $ f $ is continuous at $ x = 0 $, then the value of $ a + b $ is equal to :

If ${}^{20}C_r$ is the co-efficient of $x^r$ in the expansion of $(1+x)^{20$, then the value of $\sum_{r=0}^{20} r^2 \cdot {}^{20}C_r$ is equal to :}

Let $ Z $ be the set of all integers,
$ A = \{(x, y) \in Z \times Z : (x - 2)^2 + y^2 \leq 4\} $,
$ B = \{(x, y) \in Z \times Z : x^2 + y^2 \leq 4\} $ and
$ C = \{(x, y) \in Z \times Z : (x - 2)^2 + (y - 2)^2 \leq 4\} $
If the total number of relations from $ A \cap B $ to $ A \cap C $ is $ 2^p $, then the value of $ p $ is :

A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, -1, 1), then the projection of $\vec{OP}$ on this plane is of length :

Let M and m respectively be the maximum and minimum values of the function $f(x) = \tan^{-1}(\sin x + \cos x)$ in $[0, \pi/2]$. Then the value of $\tan(M-m)$ is equal to :

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2\vec{a} + 3\vec{b}| = |3\vec{a} + \vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$. If $\frac{1}{8} \vec{a}$ is a unit vector, then $|\vec{b}|$ is equal to :

Let A + 2B = [1 2 0; 6 -3 3; -5 3 1] and 2A - B = [2 -1 5; 2 -1 6; 0 1 2]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) - Tr(B) has value equal to :

Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

Let $f:[0, \infty) \to [0, \infty)$ be defined as $f(x) = \int_0^x [y] dy$, where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true ?