List of top Questions asked in JEE Main

A square loop of side 15 cm being moved towards right at a constant speed of 2 cm/s as shown in figure. The front edge enters the 50 cm wide magnetic field at t = 0. The value of induced emf in the loop at t = 10 s will be :

A hydrogen atom in ground state is given an energy of 10.2 eV. How many spectral lines will be emitted due to transition of electrons ?

The temperature of a gas is $-78^\circ\text{C}$ and the average translational kinetic energy of its molecules is $ K $. The temperature at which the average translational kinetic energy of the molecules of the same gas becomes $ 2K $ is:

The following figure represents two biconvex lenses $ L_1 $ and $ L_2 $ having focal lengths 10 cm and 15 cm, respectively. The distance between $ L_1 $ and $ L_2 $ is:

Consider the matrices: \[ A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, \quad B = \begin{bmatrix} 20 \\ m \end{bmatrix}, \quad \text{and} \quad X = \begin{bmatrix} x \\ y \end{bmatrix}. \] Let the set of all $ m $, for which the system of equations $ AX = B $ has a negative solution (i.e., $ x < 0 $ and $ y <0 $), be the interval $ (a, b) $. Then \[ 8 \int_a^b |\det(A)| \, dm \] is equal to _____ .

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \[ 2 \sin^{-1} x + 3 \cos^{-1} x = \frac{2\pi}{5}, \] is ______ .

The square of the distance of the image of the point $ (6, 1, 5) $ in the line \[ \frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}, \] from the origin is _____ .

Let \[ A = \{(x, y) : 2x + 3y = 23, \, x, y \in \mathbb{N}\} \] and \[ B = \{x : (x, y) \in A\}. \] Then the number of one-one functions from $ A $ to $ B $ is equal to _________ .

Let the set of all values of $ p $, for which \[ f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7 \] does not have any critical point, be the interval $ (a, b) $. Then $ 16ab $ is equal to _____ .

Let \[ \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \quad \vec{b} = -\hat{i} + \hat{k}, \quad \vec{c} = \beta \hat{j} - \hat{k}, \] where $ \alpha $ and $ \beta $ are integers and $ \alpha \beta = -6 $. Let the values of the ordered pair $ (\alpha, \beta) $ for which the area of the parallelogram of diagonals $ \vec{a} + \vec{b} $ and $ \vec{b} + \vec{c} $ is $ \frac{\sqrt{21}}{2} $, be $ (\alpha_1, \beta_1) $ and $ (\alpha_2, \beta_2) $. Then $ \alpha_1^2 + \beta_1^2 - \alpha_2 \beta_2 $ is equal to:

Let $ \alpha, \beta; \, \alpha > \beta $, be the roots of the equation $$ x^2 - \sqrt{2}x - \sqrt{3} = 0. $$ Let $ P_n = \alpha^n - \beta^n, \, n \in \mathbb{N} $. Then $$ \left( 11\sqrt{3} - 10\sqrt{2} \right) P_{10} + \left( 11\sqrt{2} + 10 \right) P_{11} - 11P_{12} $$ is equal to:

The value of the integral \[ \int_{-1}^{2} \log_e \left( x + \sqrt{x^2 + 1} \right) \, dx \] is:

Let $ a, ar, ar^2, \dots $ be an infinite G.P. If $$ \sum_{n=0}^\infty ar^n = 57 \quad \text{and} \quad \sum_{n=0}^\infty a^3 r^{3n} = 9747, $$ then $ a + 18r $ is equal to:

The integral \[ \int_{1/4}^{3/4} \cos\left( 2 \cot^{-1} \sqrt{\frac{1 - x}{1 + x}} \right) \, dx \] is equal to:

Let $$ B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix} $$ and $A$ be a $2 \times 2$ matrix such that $$ AB^{-1} = A^{-1}. $$ If $BCB^{-1} = A$ and $$ C^4 + \alpha C^2 + \beta I = O, $$ then $2\beta - \alpha$ is equal to:

\[ \lim_{x \to \frac{\pi}{2}} \frac{\int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \left( \sin\left(2t^{1/3}\right) + \cos\left(t^{1/3}\right) \right) \, dt}{\left( x - \frac{\pi}{2} \right)^2} \] is equal to:

Between the following two statements: {Statement-I:} Let \[ \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \quad \text{and} \quad \vec{b} = 2\hat{i} + \hat{j} - \hat{k}. \] Then the vector $ \vec{r} $ satisfying \[ \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \quad \text{and} \quad \vec{a} \cdot \vec{r} = 0 \] is of magnitude $ \sqrt{10} $. {Statement-II:} In a triangle $ \triangle ABC $, \[ \cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}. \]

If the variance of the frequency distribution is 160, then the value of $ c \in \mathbb{N} $ is: \[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & c & 2c & 3c & 4c & 5c & 6c \\ \hline f & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{array} \]

Two vertices of a triangle $ \triangle ABC $ are $ A(3, -1) $ and $ B(-2, 3) $, and its orthocentre is $ P(1, 1) $. If the coordinates of the point $ C $ are $ (\alpha, \beta) $ and the centre of the circle circumscribing the triangle $ \triangle PAB $ is $ (h, k) $, then the value of \[ (\alpha + \beta) + 2(h + k) \] equals:

Let \[ \int_{0}^{x} \sqrt{1 - (y'(t))^2} \, dt = \int_{0}^{x} y(t) \, dt, \quad 0 \leq x \leq 3, \, y \geq 0, \, y(0) = 0. \] Then, at $ x = 2 $, $ y'' + y + 1 $ is equal to:

$\lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{2x}}}{x} \quad \text{is equal to:}$

Number of alkanes obtained on electrolysis of a mixture of $CH_3COONa$ and $C_2H_5COONa$ is___.