List of top Questions asked in JEE Main- 2021

If $(\sin^{-1} x)^2 - (\cos^{-1} x)^2 = a; 0<x<1, a \neq 0$, then the value of $2x^2 - 1$ is :
$\int_{6}^{16} \frac{\log_e x^2}{\log_e x^2 + \log_e(x^2 - 44x + 484)} \, dx$ is equal to :
If $0<x<1$, then $\frac{3}{2}x^2 + \frac{5}{3}x^3 + \frac{7}{4}x^4 + \dots$, is equal to :
The statement $(p \land (p \to q) \land (q \to r)) \to r$ is :
$\sum_{k=0}^{20} (^{20}C_k)^2$ is equal to :
If $A = \{x \in \mathbb{R} : |x-2|>1\}$, $B = \{x \in \mathbb{R} : \sqrt{x^2-3}>1\}$, $C = \{x \in \mathbb{R} : |x-4| \geq 2\}$ and $\mathbb{Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^c \cap \mathbb{Z}$ is _________.
If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions, then $\alpha + \beta - \alpha\beta$ is equal to _________.
A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is _________.
If $y^{1/4} + y^{-1/4} = 2x$, and $(x^2 - 1)\frac{d^2y}{dx^2} + \alpha x \frac{dy}{dx} + \beta y = 0$, then $|\alpha - \beta|$ is equal to _________.
The number of distinct real roots of the equation $3x^4 + 4x^3 - 12x^2 + 4 = 0$ is _________.
Let the equation $x^2 + y^2 + px + (1 - p)y + 5 = 0$ represent circles of varying radius $r \in (0, 5]$. Then the number of elements in the set $S = \{q : q = p^2 \text{ and } q \text{ is an integer}\}$ is _________.
If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^2}{b^2} + \frac{y^2}{4a^2} = 1$ and the co-ordinate axis is kab, then k is equal to _________.
Let n be an odd natural number such that the variance of 1, 2, 3, 4, ..., n is 14. Then n is equal to _________.
Let $\vec{a} = \hat{i} + 5\hat{j} + \alpha\hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} + \beta\hat{k}$ and $\vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}$ be three vectors such that, $|\vec{b} \times \vec{c}| = 5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}$. Then the greatest amongst the values of $|\vec{a}|^2$ is _________.
If $\int \frac{dx}{(x^2 + x + 1)^2} = a \tan^{-1} \left( \frac{2x + 1}{\sqrt{3}} \right) + b \left( \frac{2x + 1}{x^2 + x + 1} \right) + C, x>0$ where C is the constant of integration, then the value of $9(\sqrt{3}a + b)$ is equal to _________.
Water drops are falling from a nozzle of a shower onto the floor, from a height of 9.8 m. The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of second drop from the floor when the first drop strikes the floor.

Match List - I with List - II. 

Choose the most appropriate answer from the options given below : 
 

Two discs have moments of inertia I\(_1\) and I\(_2\) about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, \(\omega_1\) and \(\omega_2\) respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:
A player kicks a football with an initial speed of 25 ms\(^{-1}\) at an angle of 45\(^{\circ}\) from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion? (Take g = 10 ms\(^{-2}\))

The boxes of masses 2 kg and 8 kg are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass 8 kg to strike the ground starting from rest. (use g = 10 m/s\(^2\)): 

 

A mass of 50 kg is placed at the centre of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the centre is V. The value of V is:
The height of victoria falls is 63 m. What is the difference in temperature of water at the top and at the bottom of fall?
[Given 1 cal = 4.2 J and specific heat of water = 1 cal g\(^{-1}\) \(^{\circ}\)C\(^{-1}\)]
If the rms speed of oxygen molecules at 0\(^{\circ}\)C is 160 m/s, find the rms speed of hydrogen molecules at 0\(^{\circ}\)C.

Three capacitors C\(_1\) = 2 \(\mu\)F, C\(_2\) = 6 \(\mu\)F and C\(_3\) = 12 \(\mu\)F are connected as shown in figure. Find the ratio of the charges on capacitors C\(_1\), C\(_2\) and C\(_3\) respectively: 

 

Figure shows a rod AB, which is bent in a 120\(^{\circ}\) circular arc of radius R. A charge (\(-Q\)) is uniformly distributed over rod AB. What is the electric field \(\vec{E}\) at the centre of curvature O?