List of top Questions asked in JEE Main- 2021

If the truth value of the Boolean expression $((p \vee q) \wedge (q \to r) \wedge (\sim r)) \to (p \wedge q)$ is false, then the truth values of the statements $p, q, r$ respectively can be :

The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If $\alpha$ and $\sqrt{\beta}$ are the mean and standard deviation respectively for correct data, then $(\alpha, \beta)$ is :

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is _________.

Let $ a, b \in \mathbb{R}, b \neq 0 $. Define a function \[ f(x) = \begin{cases} a \sin \frac{\pi}{2}(x - 1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases} \] If $ f $ is continuous at $ x = 0 $, then $ 10 - ab $ is equal to ________.

The sum of all integral values of $k$ ($k \neq 0$) for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is _________.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left( \frac{4}{\pi} + 1 \right)k$ is equal to _________.

The area of the region $S = \{(x, y) : 3x^2 \leq 4y \leq 6x + 24\}$ is _________.

If $y = y(x)$ is an implicit function of $x$ such that $\log_e (x + y) = 4xy$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is equal to _________.

If ${}^1P_1 + 2 \cdot {}^2P_2 + 3 \cdot {}^3P_3 + \dots + 15 \cdot {}^{15}P_{15} = {}^qP_r - s$, where $0 \leq s \leq 1$, then $q + s + {}^qC_{r - s}$ is equal to _________.

Let the line $L$ be the projection of the line \[ \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2} \] in the plane $x - 2y - z = 3$. If $d$ is the distance of the point $(0, 0, 6)$ from $L$, then $d^2$ is equal to _________.

Let $ z = \frac{1 - i\sqrt{3}}{2}, \, i = \sqrt{-1} $. Then the value of

\[ 21 + \left( z + \frac{1}{z} \right)^3 + \left( z^2 + \frac{1}{z^2} \right)^3 + \left( z^3 + \frac{1}{z^3} \right)^3 + \dots + \left( z^{21} + \frac{1}{z^{21}} \right)^3 \] is _________.

The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter $d$. Then $d^2$ is equal to _________.

Moment of inertia of a square plate of side l about the axis passing through one of the corner and perpendicular to the plane of square plate is given by :

The resultant of these forces $\vec{OP}, \vec{OQ}, \vec{OR}, \vec{OS}$ and $\vec{OT}$ is approximately ________ N.
[Take $\sqrt{3} = 1.7, \sqrt{2} = 1.4$. Given $\hat{i}$ and $\hat{j}$ unit vectors along x, y axis]

If E and H represents the intensity of electric field and magnetising field respectively, then the unit of E/H will be :

An ideal gas is expanding such that $PT^3 = \text{constant}$. The coefficient of volume expansion of the gas is :

A balloon carries a total load of 185 kg at normal pressure and temperature of 27°C. What load will the balloon carry on rising to a height at which the barometric pressure is 45 cm of Hg and the temperature is -7°C. Assuming the volume constant ?

The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure.
The potential energy U(x) versus time (t) plot of the particle is correctly shown in figure :

Calculate the amount of charge on capacitor of 4 $\mu$F. The internal resistance of battery is 1 $\Omega$:

A uniformly charged disc of radius R having surface charge density $\sigma$ is placed in the xy plane with its center at the origin. Find the electric field intensity along the z-axis at a distance Z from origin :

Two ions of masses 4 amu and 16 amu have charges +2e and +3e respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then :

A bar magnet is passing through a conducting loop of radius R with velocity v. The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve :

Electric field in a plane electromagnetic wave is given by E = 50 sin(500x - 10 $\times$ 10$^{10}$t) V/m. The velocity of electromagnetic wave in this medium is : (Given C = speed of light in vacuum)

An object is placed beyond the centre of curvature C of the given concave mirror. If the distance of the object is $d_1$ from C and the distance of the image formed is $d_2$ from C, the radius of curvature of this mirror is :

In a photoelectric experiment, increasing the intensity of incident light :