List of top Questions asked in JEE Main- 2021

If the curve $x^2+2y^2 = 2$ intersects the line $x+y=1$ at two points P and Q, then the angle subtended by the line segment PQ at the origin is :
The shortest distance between the line $x-y=1$ and the curve $x^2 = 2y$ is:
A hyperbola passes through the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola 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 :
$\lim_{n\to\infty} [\frac{1}{n} + \frac{n}{(n+1)^2} + \frac{n}{(n+2)^2} + \dots + \frac{n}{(2n-1)^2}]$ is equal to :
In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:
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 :
If $0<x, y<\pi$ and $\cos x + \cos y - \cos(x+y) = \frac{3}{2}$, then $\sin x + \sin y$ is equal to :
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 :
cosec[2cot$^{-1}$(5) + cos$^{-1}$($\frac{4}{5}$)] is equal to :
The contrapositive of the statement "If you will work, you will earn money" is:
A function \( f \) is defined on \([-3, 3]\) as \( f(x) = \begin{cases} \min\{|x|, 2-x^2\}, & -2 \le x \le 2 \\ |x|, & 2 < |x| \le 3 \end{cases} \), where \( [x] \) denotes the greatest integer \( \le x \). The number of points, where \( f \) is not differentiable in \((-3, 3)\) is ________ .
If the curve, y=y(x) represented by the solution of the differential equation $(2xy^2-y)dx+xdy=0$, passes through the intersection of the lines, 2x$-$3y=1 and 3x+2y=8, then $|y(1)|$ is equal to ________ .
The total number of two digit numbers 'n', such that $3^n + 7^n$ is a multiple of 10, is ________ .
If lim$_{x\to 0} \frac{ax - (e^{4x}-1)}{ax(e^{4x}-1)}$ exists and is equal to b, then the value of a$-$2b is ________ .
If the curves $x=y^4$ and $xy=k$ cut at right angles, then $(4k)^6$ is equal to ________ .
The value of $\int_{-2}^{2} |3x^2-3x-6| \,dx$ is ________ .
If the remainder when x is divided by 4 is 3, then the remainder when $(2020+x)^{2022}$ is divided by 8 is ________ .
A line 'l' passing through origin is perpendicular to the lines
$l_1: \vec{r} = (3+t)\hat{i} + (-1+2t)\hat{j} + (4+2t)\hat{k}$
$l_2: \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}$
If the co-ordinates of the point in the first octant on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of 'l' and '$l_1$' are (a, b, c), then 18(a+b+c) is equal to ________ .
A line is a common tangent to the circle $(x-3)^2+y^2=9$ and the parabola $y^2=4x$. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a+c) is equal to ________ .
Let $\vec{a} = \hat{i} + \alpha\hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} - \alpha\hat{j} + \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is 8$\sqrt{3}$ square units, then $\vec{a} \cdot \vec{b}$ is equal to ________ .
If 'f' denotes the ratio of the number of nuclei decayed Nd to the number of nuclei at t=0 N0 then for a collection of radioactive nuclei, the rate of change of 'f' with respect to time is given as: [$\lambda$ is the radioactive decay constant]
In Young's double slit experiment, if the source of light changes from orange to blue then:
The relative permittivity of distilled water is 81. The velocity of light in it will be : (Given $\mu_r = 1$)

A capacitor of capacitance C=1 $\mu$F is suddenly connected to a battery of 100 volt through a resistor R = 100 $\Omega$. The time taken for the capacitor to be charged to get 50 V is: [Take ln 2 = 0.69]