List of top Questions asked in JEE Main- 2021

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 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 :
Let $f: (a,b) \to \mathbb{R}$ be twice differentiable function such that $f(x) = \int_a^x g(t)dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g'(x)=0$ has at least :
Let A and B be two $3 \times 3$ real matrices such that $(A^2 - B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3B^2=A^2B^3$, then the value of the determinant of the matrix $A^3+B^3$ is equal to :
Let $\mathbb{N}$ be the set of natural numbers and a relation R on $\mathbb{N}$ be defined by R = $\{(x, y) \in \mathbb{N} \times \mathbb{N} : x^3 - 3x^2y - xy^2 + 3y^3 = 0\}$. Then the relation R is :
Let $\alpha = \max_{x \in \mathbb{R}}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$ and $\beta = \min_{x \in \mathbb{R}}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$.
If $8x^2+bx+c=0$ is a quadratic equation whose roots are $\alpha^{1/5}$ and $\beta^{1/5}$, then the value of c-b is equal to :
The value of $\lim_{x \to 0} \left(\frac{x}{\sqrt[8]{1-\sin x} - \sqrt[8]{1+\sin x}}\right)$ is equal to :
A student appeared in an examination consisting of 8 true - false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least 'n' correct answers is less than $\frac{1}{2}$, is :
For real numbers $\alpha$ and $\beta \neq 0$, if the point of intersection of the straight lines
$\frac{x-\alpha}{1} = \frac{y-1}{2} = \frac{z-1}{3}$ and $\frac{x-4}{\beta} = \frac{y-6}{3} = \frac{z-7}{3}$
lies on the plane $x+2y-z=8$, then $\alpha-\beta$ is equal to :
If $\tan(\frac{\pi}{9})$, x, $\tan(\frac{7\pi}{18})$ are in arithmetic progression and $\tan(\frac{\pi}{9})$, y, $\tan(\frac{5\pi}{18})$ are also in arithmetic progression, then $|x-2y|$ is equal to :
Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a} = \vec{b} \times (\vec{b} \times \vec{c})$. If magnitudes of the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta$ ($0<\theta<\frac{\pi}{2}$), then the value of $1+\tan\theta$ is equal to :
If the real part of the complex number $z = \frac{3+2i\cos\theta}{1-3i\cos\theta}$, $\theta \in (0, \frac{\pi}{2})$ is zero, then the value of $\sin^2 3\theta + \cos^2 \theta$ is equal to ________.
If \( A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \) and \( M = A + A^2 + A^3 + \cdots + A^{20} \), then the sum of all the elements of the matrix \( M \) is equal to ________.
Let n be a non-negative integer. Then the number of divisors of the form "4n+1" of the number $(10)^{10} \cdot (11)^{11} \cdot (13)^{13}$ is equal to ________.
The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points Q(3, -4, -5) and R(2, -3, 1) and the plane $2x+y+z=7$, is equal to ________.
Let $y=y(x)$ be the solution of the differential equation $dy = e^{\alpha x+y}dx; \alpha \in \mathbb{N}$. If $y(\log_e 2) = \log_e 2$ and $y(0)=\log_e(\frac{1}{2})$, then the value of $\alpha$ is equal to ________.
If $\int_0^\pi (\sin^3 x) e^{-\sin^2 x} dx = \alpha - \frac{\beta}{e} \int_0^1 \sqrt{t} e^t dt$, then $\alpha+\beta$ is equal to ________.
Let $\vec{a}=\hat{i}-\alpha\hat{j}+\beta\hat{k}$, $\vec{b}=3\hat{i}+\beta\hat{j}-\alpha\hat{k}$ and $\vec{c}=-\alpha\hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b} = -1$ and $\vec{b} \cdot \vec{c} = 10$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to ________.
Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, -4), one focus at (4, -4) and one vertex at (5, -4). If $mx-y=4, m>0$ is a tangent to the ellipse E, then the value of $5m^2$ is equal to ________.
Let $A = \{n \in \mathbb{N} | n^2 \le n+10000\}$, $B = \{3k+1 | k \in \mathbb{N}\}$ and $C = \{2k | k \in \mathbb{N}\}$. Then the sum of all the elements of the set $A \cap (B-C)$ is equal to ________.
The number of real roots of the equation $e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$ is equal to ________.
The half-life of $^{198}\text{Au}$ is 3 days. If atomic weight of $^{198}\text{Au}$ is 198 g/mol then the activity of 2 mg of $^{198}\text{Au}$ is [in disintegration/second] :
A parallel plate capacitor with plate area 'A' and distance of separation 'd' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as : \[ \epsilon(x) = \epsilon_0 + kx, \text{ for } \left( 0<x \le \frac{d}{2} \right) \] \[ \epsilon(x) = \epsilon_0 + k(d-x), \text{ for } \left( \frac{d}{2} \le x \le d \right) \]
Water droplets are coming from an open tap at a particular rate. The spacing between a droplet observed at 4th second after its fall to the next droplet is 34.3 m. At what rate the droplets are coming from the tap? (Take $g = 9.8 \text{ m/s}^2$)
In Amplitude Modulation, the message signal $V_m(t) = 10 \sin(2\pi \times 10^5 t)$ volts and Carrier signal $V_c(t) = 20 \sin(2\pi \times 10^7 t)$ volts. The modulated signal now contains the message signal with lower side band and upper side band frequency, therefore the bandwidth of modulated signal is $\alpha \text{ kHz}$. The value of $\alpha$ is :