List of top Questions asked in JEE Main

A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude \(\theta_0\). If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his center of mass moves by a distance \(\ell\) (\(\ell\ \lt\lt\ L\)), is close to;

An electromagnetic wave is represented by the electric field
\(\vec{E}=E_0\hat{n}\sin\left[\omega t+(6y-8z)\right]\)
. Taking unit vectors in x, y and z directions to be \(\hat{i},\hat{j},\hat{k}\), the direction of propagation \(\hat{s}\), is :

An excited He\(^+\) ion emits two photons in succession, with wavelengths 108.5 nm and 30.4 nm, in making a transition to ground state. The quantum number n, corresponding to its initial excited state is (for photon of wavelength \(\lambda\), energy
\(E=\frac{1240\,\mathrm{eV}}{\lambda\,(\text{in nm})}\)
) :

Let a random variable X have a binomial distribution with mean 8 and variance 4. If
\(P(X\le2)=\frac{k}{2^{16}}\)
, then k is equal to :

For \(x\in\mathbb{R}\), let [x] denote the greatest integer \(\le x\), then the sum of the series
\(\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\ldots+\left[-\frac{1}{3}-\frac{99}{100}\right]\)
is :

If the line
\(\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}\)
intersects the plane \(2x+3y-z+13=0\) at a point P and the plane \(3x+y+4z=16\) at a point Q, then PQ is equal to :

The number of solutions of the equation
\(1+\sin^{4}x=\cos^{2}3x\),
\(x\in\left[-\frac{5\pi}{2},\frac{5\pi}{2}\right]\)
is :

Let P be the point of intersection of the common tangents to the parabola \(y^2=12x\) and the hyperbola \(8x^2-y^2=8\). If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :

If the normal to the ellipse \(3x^2+4y^2=12\) at a point P on it is parallel to the line, \(2x+y=4\) and the tangent to the ellipse at P passes through Q(4,4) then PQ is equal to :

If the volume of parallelopiped formed by the vectors \(\hat{i}+\lambda\hat{j}+\hat{k}\), \(\hat{j}+\lambda\hat{k}\) and \(\lambda\hat{i}+\hat{k}\) is minimum, then \(\lambda\) is equal to :

The equation \(y=\sin x\,\sin(x+2)-\sin^{2}(x+1)\) represents a straight line lying in :

For \(x \in (0, 3/2)\), let \(f(x) = \sqrt{x}\), \(g(x) = \tan x\) and \(h(x) = \frac{1 - x^2}{1 + x^2}\). If \(\phi(x) = ((h\circ f)\circ g)(x)\), then \(\phi\left(\frac{\pi}{3}\right)\) is equal to :

If the area (in sq. units) of the region \(\{(x, y) : y^2 \le 4x,\ x + y \le 1,\ x \ge 0,\ y \ge 0\}\) is \(a\sqrt{2} + b\), then \(a - b\) is equal to :

Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuously differentiable function such that \(f(2)=6\) and \(f'(2)=\frac{1}{48}\). If
\(\int_{6}^{f(x)} 4t^{3}\,dt=(x-2)\,g(x)\),
then \(\lim_{x\to 2} g(x)\) is equal to :

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is :

If \(e^y + xy = e\), the ordered pair \(\left(\frac{dy}{dx}, \frac{d^2y}{dx^2}\right)\) at \(x = 0\) is equal to :

The coefficient of \(x^{18}\) in the product \((1 + x)(1 - x)^{10}(1 + x + x^2)^9\) is :

The equation \(|z - i| = |z - 1|\), \(i = \sqrt{-1}\), represents :

If \(\alpha\) and \(\beta\) are the roots of the equation \(375x^2 - 25x - 2 = 0\), then \(\lim_{n\to\infty}\sum_{r=1}^{n}\alpha^r + \lim_{n\to\infty}\sum_{r=1}^{n}\beta^r\) is equal to :

If the data \(x_1, x_2, \ldots\ldots, x_{10}\) is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is :

If m is the minimum value of k for which the function \(f(x) = x\sqrt{kx - x^2}\) is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :

If \(B = \begin{bmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{bmatrix}\) is the inverse of a \(3 \times 3\) matrix A, then the sum of all values of \(\alpha\) for which det(A) + 1 = 0, is :

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = \(\begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}\), then AB is equal to :

The value of \(\sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right)\) is equal to :