List of top Questions asked in JEE Main- 2021

Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :
The number of solutions of the equation \(32^{\tan^2 x} + 32^{\sec^2 x} = 81\), \(0 \le x \le \frac{\pi}{4}\) is :
Let \(*, \square \in \{\wedge, \vee\}\) be such that the Boolean expression \((p * \sim q) \implies (p \square q)\) is a tautology. Then :
Let C be the set of all complex numbers. Let
$S_1 = \{z \in C : |z-2| \le 1\}$ and
$S_2 = \{z \in C : z(1+i) + \bar{z}(1-i) \ge 4\}$.
Then, the maximum value of $|z-\frac{5}{2}|^2$ for $z \in S_1 \cap S_2$ is equal to :
An angle of intersection of the curves, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and \( x^2 + y^2 = ab, a>b \), is :
Let \( f \) be any continuous function on \( [0, 2] \) and twice differentiable on \( (0, 2) \). If \( f(0) = 0, f(1) = 1 \) and \( f(2) = 2 \), then :
Let \(A\) and \(B\) be independent events such that \(P(A) = p, P(B) = 2p\). The largest value of \(p\), for which \(P(\text{exactly one of } A, B \text{ occurs}) = \frac{5{9}\), is :}
The area (in sq. units) of the region, given by the set $\{(x, y) \in R \times R \mid x \ge 0, 2x^2 \le y \le 4 - 2x\}$ is :
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set\{P, Q\} is equal to :
\(\csc 18^\circ\) is a root of the equation :
Two tangents are drawn from the point P(-1, 1) to the circle $x^2 + y^2 - 2x - 6y + 6 = 0$. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :
Let 9 distinct balls be distributed among 4 boxes, $B_1, B_2, B_3$ and $B_4$. If the probability that $B_3$ contains exactly 3 balls is $k \left(\frac{3}{4}\right)^9$ then $k$ lies in the set :
In a triangle ABC, if |BC| = 8, |CA| = 7, |AB| = 10, then the projection of the vector AB on AC is equal to :
The sum of the series \(\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \dots + \frac{2^{100}}{x^{2^{100}}+1}\) when \(x = 2\) is :
If \( 0 < \theta, \phi < \frac{\pi}{2} \), \( x = \sum_{n=0}^{\infty} \cos^{2n} \theta \), \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \) and \( z = \sum_{n=0}^{\infty} \cos^{2n} \theta \cdot \sin^{2n} \phi \) then :
If \( y \frac{dy}{dx} = x \left[ \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} + \frac{y^2}{x^2} \right], x>0, \phi>0, \) and \( y(1) = -1 \), then \( \phi\left(\frac{y^2}{4}\right) \) is equal to :
The equation of the line through the point (0, 1, 2) and perpendicular to the line (x-1)/2 = (y+1)/3 = (z-1)/(-2) is:
If the matrix \( A = \begin{pmatrix} 0 & 2 \\ K & -1 \end{pmatrix} \) satisfies \( A(A^3 + 3I) = 2I \), then the value of \( K \) is :
The Boolean expression $(p \implies q) \wedge (q \implies \sim p)$ is equivalent to :
Define a relation R over a class of n × n real matrices A and B as "ARB iff there exists a non-singular matrix P such that P A P⁻¹ = B". Then which of the following is true ?
Let the mean and variance of the frequency distribution be 6 and 6.8 respectively.
x : \( x_1 = 2,\; x_2 = 6,\; x_3 = 8,\; x_4 = 9 \)
f : \( 4,\; 4,\; \alpha,\; \beta \)
If \( x_3 \) is changed from 8 to 7, then the mean for the new data will be:
The integer k, for which the inequality x² - 2(3k-1)x + 8k² - 7>0 is valid for every x in ℝ, is :
For p and q, consider: (a) (~ q ∧ (p → q)) → ~ p, (b) ((p ∨ q) ∧ ~ p) → q. Which is correct ?
The differential equation satisfied by the system of parabolas y² = 4a(x + a) is :
The population $P = P(t)$ at time 't' of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$, then the time at which population becomes zero is :