List of top Questions asked in JEE Main- 2021

A butterfly is flying with a velocity \(4\sqrt{2}\) m/s in North-East direction. Wind is slowly blowing at 1 m/s from North to South. The resultant displacement of the butterfly in 3 seconds is :

A deuteron and an alpha particle having equal kinetic energy enter perpendicularly into a magnetic field. Let \(r_d\) and \(r_\alpha\) be their respective radii of circular path. The value of \(\frac{r_d}{r_\alpha}\) is equal to :

Let \(y = mx + c\), \(m \gt 0\) be the focal chord of \(y^2 = -64x\), which is tangent to \((x + 10)^2 + y^2 = 4\). Then, the value of \(4\sqrt{2}\,(m + c)\) is equal to _____________.

A current of 5 A is passing through a non-linear magnesium wire of cross-section \(0.04\ \mathrm{m^2}\). At every point the direction of current density is at an angle of \(60^\circ\) with the unit vector of area of cross-section. The magnitude of electric field at every point of the conductor is :

(Resistivity of magnesium \(\rho = 44 \times 10^{-8}\ \Omega\mathrm{m}\))

For the circuit shown below, calculate the value of \(I_z\) :

Let P be a plane passing through the points (1, 0, 1), (1, -2, 1) and (0, 1, -2). Let a vector \(\vec{a} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}\) be such that \(\vec{a}\) is parallel to the plane P, perpendicular to \((\hat{i} + 2\hat{j} + 3\hat{k})\) and \(\vec{a}\cdot(\hat{i} + \hat{j} + 2\hat{k}) = 2\), then \((\alpha - \beta + \gamma)^2\) equals ____________.

If the shortest distance between the lines \(\vec{r_1} = \alpha\hat{i} + 2\hat{j} + 2\hat{k} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})\), \(\lambda \in \mathbb{R}\), \(\alpha \gt 0\) and \(\vec{r_2} = -4\hat{i} - \hat{k} + \mu(3\hat{i} - 2\hat{j} - 2\hat{k})\), \(\mu \in \mathbb{R}\) is 9, then \(\alpha\) is equal to ____________.

Let \(A = \begin{pmatrix}1 & -1 & 0\\0 & 1 & -1\\0 & 0 & 1\end{pmatrix}\) and \(B = 7A^{20} - 20A^{7} + 2I\), where I is an identity matrix of order \(3 \times 3\). If \(B = [b_{ij}]\), then \(b_{13}\) is equal to _____________.

The number of rational terms in the binomial expansion of \(\left(4^{\frac{1}{4}} + 5^{\frac{1}{6}}\right)^{120}\) is _______________.

If the value of \(\lim_{x \to 0}\left(2 - \cos x\sqrt{\cos 2x}\right)^{\left(\frac{x+2}{x^2}\right)}\) is equal to \(e^a\), then a is equal to __________.

Let T be the tangent to the ellipse E : \(x^2 + 4y^2 = 5\) at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines \(x = 1\) and \(x = \sqrt{5}\) is \(\alpha\sqrt{5} + \beta + \gamma\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\), then \(|\alpha + \beta + \gamma|\) is equal to ______________.

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsman and 1 wicketkeeper, is ______________.

Let \(\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}\) and \(\vec{b} = \hat{i} + \hat{j}\). If \(\vec{c}\) is a vector such that \(\vec{a}\cdot\vec{c} = |\vec{c}|,\ |\vec{c} - \vec{a}| = 2\sqrt{2}\) and the angle between \((\vec{a}\times\vec{b})\) and \(\vec{c}\) is \(\frac{\pi}{6}\), then the value of \(|(\vec{a}\times\vec{b})\times\vec{c}|\) is :

The coefficient of \(x^{256}\) in the expansion of
\((1 - x)^{101}(x^2 + x + 1)^{100}\) is :

If in a triangle ABC, AB = 5 units, \(\angle B = \cos^{-1}\left(\frac{3}{5}\right)\) and radius of circumcircle of \(\Delta\)ABC is 5 units, then the area (in sq. units) of \(\Delta\)ABC is :

Let the tangent to the parabola S : \(y^2 = 2x\) at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

Let \(\vec{a}, \vec{b}, \vec{c}\) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \(\theta\), with the vector \(\vec{a} + \vec{b} + \vec{c}\). Then \(36\cos^2 2\theta\) is equal to ___________.

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :

Let a function \(f:\mathbb{R}\to\mathbb{R}\) be defined as
\(f(x)=\left\{\begin{matrix}\sin x-e^x&\text{if}&x\le0\\a+[-x]&\text{if}&0\lt x\lt1\\2x-b&\text{if}&x\ge1\end{matrix}\right.\)
where \([x]\) is the greatest integer less than or equal to \(x\). If \(f\) is continuous on \(\mathbb{R}\), then \((a+b)\) is equal to:

The Boolean expression \((p \wedge \sim q) \Rightarrow (q \vee \sim p)\) is equivalent to :

The probability of selecting integers \(a\in[-5,30]\) such that \(x^2+2(a+4)x-5a+64\gt0\), for all \(x\in\mathbb{R}\), is :

Let 'a' be a real number such that the function \(f(x)=ax^2+6x-15,\ x\in\mathbb{R}\) is increasing in
\(\left(-\infty,\frac34\right)\)
and decreasing in
\(\left(\frac34,\infty\right)\)
. Then the function \(g(x)=ax^2-6x+15,\ x\in\mathbb{R}\) has a :

The number of real roots of the equation
\(\tan^{-1}\sqrt{x(x+1)}+\sin^{-1}\sqrt{x^2+x+1}=\frac{\pi}{4}\)
is :

Let \(y=y(x)\) be the solution of the differential equation
\(e^x\sqrt{1-y^2}\,dx+\left(\frac{y}{x}\right)dy=0\)
, \(y(1)=-1\). Then the value of \((y(3))^2\) is equal to :

Let \(y=y(x)\) be the solution of the differential equation
\(x\tan\left(\frac{y}{x}\right)dy=\left(y\tan\left(\frac{y}{x}\right)-x\right)dx\)
, \(-1\le x\le1\),
\(y\left(\frac12\right)=\frac{\pi}{6}\)
. Then the area of the region bounded by the curves \(x=0\),
\(x=\frac{1}{\sqrt2}\)
and \(y=y(x)\) in the upper half plane is :