List of top Questions asked in JEE Main- 2021

What will be the projection of vector
\(\vec{A} = \hat{i} + \hat{j} + \hat{k}\)
on vector
\(\vec{B} = \hat{i} + \hat{j}\) ?

Let \(y = y(x)\) be the solution of the differential equation
\(\left((x + 2)e^{\left(\frac{y + 1}{x + 2}\right)} + (y + 1)\right)dx = (x + 2)\,dy\), \(y(1) = 1\).
If the domain of \(y = y(x)\) is an open interval \((\alpha, \beta)\), then \(|\alpha + \beta|\) is equal to _____________.

The area (in sq. units) of the region bounded by the curves \(x^2 + 2y - 1 = 0\), \(y^2 + 4x - 4 = 0\) and \(y^2 - 4x - 4 = 0\), in the upper half plane is _______________.

The number of elements in the set \(\{n \in \{1, 2, 3, \ldots\ldots, 100\} \mid (11)^n \gt (10)^n + (9)^n\}\) is ______________.

If the constant term, in binomial expansion of \(\left(2x^r + \frac{1}{x^2}\right)^{10}\) is 180, then \(r\) is equal to __________________.

Let a line \(L : 2x + y = k,\ k \gt 0\) be a tangent to the hyperbola \(x^2 - y^2 = 3\). If \(L\) is also a tangent to the parabola \(y^2 = \alpha x\), then \(\alpha\) is equal to :

The number of solutions of \(\sin^7 x + \cos^7 x = 1\), \(x \in [0, 4\pi]\) is equal to

Consider the following frequency distribution :

Class :	0-6	6-12	12-18	18-24	24-30
Frequency :	\(a\)	\(b\)	12	9	5

If mean \(= \frac{309}{22}\) and median \(= 14\), then the value \((a - b)^2\) is equal to _____________.

Let \(A = \{0, 1, 2, 3, 4, 5, 6, 7\}\). Then the number of bijective functions \(f : A \to A\) such that \(f(1) + f(2) = 3 - f(3)\) is equal to

Let \(A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). Then the number of \(3 \times 3\) matrices \(B\) with entries from the set \(\{1, 2, 3, 4, 5\}\) and satisfying \(AB = BA\) is ____________.

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to _____________.

Let three vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be such that \(\vec{a}\times\vec{b} = \vec{c}\), \(\vec{b}\times\vec{c} = \vec{a}\) and \(|\vec{a}| = 2\). Then which one of the following is not true?

Let L be the line of intersection of planes \(\vec{r}\cdot(\hat{i} - \hat{j} + 2\hat{k}) = 2\) and \(\vec{r}\cdot(2\hat{i} + \hat{j} - \hat{k}) = 2\). If \(P(\alpha, \beta, \gamma)\) is the foot of perpendicular on L from the point \((1, 2, 0)\), then the value of \(35(\alpha + \beta + \gamma)\) is equal to :

Let \(E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\ a \gt b\). Let \(E_2\) be another ellipse such that it touches the end points of major axis of \(E_1\) and the foci of \(E_2\) are the end points of minor axis of \(E_1\). If \(E_1\) and \(E_2\) have same eccentricities, then its value is :

Which of the following Boolean expressions is not a tautology?

Let the circle \(S : 36x^2 + 36y^2 - 108x + 120y + C = 0\) be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, \(x - 2y = 4\) and \(2x - y = 5\) lies inside the circle S, then :

Let \(f : R \to R\) be defined as \(f(x) = \begin{cases} \frac{x^3}{(1 - \cos 2x)^2}\log_e\left(\frac{1 + 2xe^{-2x}}{(1 - xe^{-x})^2}\right), & x \ne 0 \\ \alpha, & x = 0 \end{cases}\)
If f is continuous at \(x = 0\), then \(\alpha\) is equal to :

Let n denote the number of solutions of the equation \(z^2 + 3\overline{z} = 0\), where z is a complex number. Then the value of \(\sum_{k=0}^{\infty}\frac{1}{n^k}\) is equal to :

If the shortest distance between the straight lines \(3(x - 1) = 6(y - 2) = 2(z - 1)\) and \(4(x - 2) = 2(y - \lambda) = (z - 3),\ \lambda \in R\) is \(\frac{1}{\sqrt{38}}\), then the integral value of \(\lambda\) is equal to :

Let \(A = [a_{ij}]\) be a real matrix of order \(3 \times 3\), such that \(a_{i1} + a_{i2} + a_{i3} = 1\), for \(i = 1, 2, 3\). Then, the sum of all the entries of the matrix \(A^3\) is equal to :

Let \([x]\) denote the greatest integer less than or equal to x. Then, the values of \(x \in R\) satisfying the equation \([e^x]^2 + [e^x + 1] - 3 = 0\) lie in the interval :

The values of \(\lambda\) and \(\mu\) such that the system of equations \(x + y + z = 6\), \(3x + 5y + 5z = 26\), \(x + 2y + \lambda z = \mu\) has no solution, are :

If \(\int_0^{100\pi} \frac{\sin^2 x}{e^{\left(\frac{x}{\pi} - \left[\frac{x}{\pi}\right]\right)}}dx = \frac{\alpha\pi^3}{1 + 4\pi^2},\ \alpha \in R\) where \([x]\) is the greatest integer less than or equal to x, then the value of \(\alpha\) is :

Four dice are thrown simultaneously and the numbers shown on these dice are recorded in \(2 \times 2\) matrices. The probability that such formed matrix have all different entries and are non-singular, is :

Let \(f : R \to R\) be defined as
\(f(x) = \begin{cases} -\frac{4}{3}x^3 + 2x^2 + 3x, & x \gt 0 \\ 3xe^x, & x \le 0 \end{cases}\)
Then f is increasing function in the interval