List of top Questions asked in JEE Main

A parallel plate capacitor is made of two square plates of side 'a', separated by a distance d \((d \ll a)\). The lower triangular portion is filled with a dielectric of dielectric constant K, as shown in the figure. Capacitance of this capacitor is :

Mobility of electrons in a semiconductor is defined as the ratio of their drift velocity to the applied electric field. If, for an n-type semiconductor, the density of electrons is \(10^{19}\ \mathrm{m^{-3}}\) and their mobility is \(1.6\ \mathrm{m^2/(V.s)}\) then the resistivity of the semiconductor (since it is an n-type semiconductor contribution of holes is ignored) is close to :

A particle is moving with a velocity \(\vec{v}=K(y\hat{i}+x\hat{j})\), where K is a constant. The general equation for its path is :

A conducting circular loop made of a thin wire, has area \(3.5\times10^{-3}\ \mathrm{m^2}\) and resistance \(10\ \Omega\). It is placed perpendicular to a time dependent magnetic field \(B(t)=(0.4\,\mathrm{T})\sin(50\pi t)\). The field is uniform in space. Then the net charge flowing through the loop during \(t=0\ \mathrm{s}\) and \(t=10\ \mathrm{ms}\) is close to :

An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is a and distance of its centre from the wire is d \((d \gg a)\). If the loop a applies a force F on the wire then :

Consider a tank made of glass(refractive index 1.5) with a thick bottom. It is filled with a liquid of refractive index \(\mu\). A student finds that, irrespective of what the incident angle i (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of \(\mu\) is :

Let \(\vec{a}=\hat{i}-\hat{j}\), \(\vec{b}=\hat{i}+\hat{j}+\hat{k}\) and \(\vec{c}\) be a vector such that \(\vec{a}\times\vec{c}+\vec{b}=\vec{0}\) and \(\vec{a}\cdot\vec{c}=4\), then \(|\vec{c}|^2\) is equal to :

Consider the set of all lines \(px+qy+r=0\) such that \(3p+2q+4r=0\). Which one of the following statements is true?

A bar magnet is demagnetized by inserting it inside a solenoid of length 0.2 m, 100 turns, and carrying a current of 5.2 A. The corecivity of the bar magnet is :

If the Boolean expression \((p\oplus q)\wedge(\sim p\odot q)\) is equivalent to \(p\wedge q\), where \(\oplus,\odot\in\{\wedge,\vee\}\), then the ordered pair \((\oplus,\odot)\) is :

Let \(0 \lt \theta \lt \frac{\pi}{2}\). If the eccentricity of the hyperbola \(\frac{x^2}{\cos^2\theta}-\frac{y^2}{\sin^2\theta}=1\) is greater than 2, then the length of its latus rectum lies in the interval :

The system of linear equations
\(x + y + z = 2\)
\(2x + 3y + 2z = 5\)
\(2x + 3y + (a^2 - 1)z = a + 1\)
then

If \(\theta\) denotes the acute angle between the curves, \(y = 10 - x^2\) and \(y = 2 + x^2\) at a point of their intersection, the \(|\tan\theta|\) is equal to :

Equation of a common tangent to the circle, \(x^2+y^2-6x=0\) and the parabola, \(y^2=4x\) is :

Three circles of radii a, b, c \((a \lt b \lt c)\) touch each other externally. If they have x-axis as a common tangent, then :

The equation of the line passing through \((-4, 3, 1)\), parallel to the plane \(x+2y-z-5=0\) and intersecting the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}\) is :

5 students of a class have an average height 150 cm and variance 18 cm². A new student, whose height is 156 cm, joined them. The variance (in cm²) of the height of these six students is :

Let \(A = \left\{\theta \in \left(-\frac\pi2, \pi\right) : \frac{3 + 2i\sin\theta}{1 - 2i\sin\theta}\ \text{is purely imaginary}\right\}\). Then the sum of the elements in A is :

If \(A = \begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}\), then the matrix \(A^{-50}\) when \(\theta = \frac{\pi}{12}\), is equal to :

If \(\cos^{-1}\left(\frac{2}{3x}\right) + \cos^{-1}\left(\frac{3}{4x}\right) = \frac\pi2\) \(\left(x \gt \frac34\right)\), then \(x\) is equal to :

If \(y = y(x)\) is the solution of the differential equation,
\(x\dfrac{dy}{dx} + 2y = x^2\), satisfying \(y(1) = 1\), then \(y\left(\frac12\right)\) is equal to :

Let \(f : R \to R\) be a function defined as
\(f(x) = \begin{cases}5 & ; x \le 1\\ a + bx & ; 1 \lt x \lt 3\\ b + 5x & ; 3 \le x \lt 5\\ 30 & ; x \ge 5\end{cases}\)
Then, f is

Let \(\alpha\) and \(\beta\) be two roots of the equation \(x^2 + 2x + 2 = 0\), then \(\alpha^{15} + \beta^{15}\) is equal to :

The value of \(\displaystyle\int_0^{\pi}|\cos x|^3\,dx\) is :

\(\displaystyle\lim_{y \to 0}\frac{\sqrt{1 + \sqrt{1 + y^4}} - \sqrt2}{y^4}\)