List of top Questions asked in JEE Main

Let $ A \in \mathbb{R} $ be a matrix of order 3x3 such that $$ \det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1 \\2 & 0 & 1 \\4 & 1 & 2 \end{array} \right] $$ where $ I $ is the identity matrix of order 3. If $ \det( (A + I) \cdot \text{adj}(A + I)) $ is $ 2^m $, then $ m $ is equal to:

If the components of $ \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} $ along and perpendicular to $ \vec{b} = 3\hat{i} + \hat{j} - \hat{k} $ respectively are $ \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) $ and $ \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

If the shortest distance between the lines $ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} $ and $ \frac{x}{1} = \frac{y}{\alpha} = \frac{z-5}{1} $ is $ \frac{5}{\sqrt{6}} $, then the sum of all possible values of $ \alpha $ is:

If the domain of the function $ f(x) = \frac{1}{\sqrt{3x + 10 - x^2}} + \frac{1}{\sqrt{x + |x|}} $ is $ (a, b) $, then $ (1 + a)^2 + b^2 $ is equal to:

Let $ A $ be a $ 3 \times 3 $ matrix such that $ X^T AX = 0 $ for all nonzero $ 3 \times 1 $ matrices $ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $. \[ A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] If $ \det(\text{adj}(2(A + I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} $, $ \alpha, \beta, \gamma \in \mathbb{N} $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is:

In the given circuit the sliding contact is pulled outwards such that the electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 Ω, the value of the current in the circuit will be A.

Let $ f: [0, 3] \to A $ be defined by $ f(x) = 2x^3 - 15x^2 + 36x + 7 $ and $ g: [0, \infty) \to B $ be defined by $ g(x) = \frac{x}{x^{2025} + 1}. $ If both functions are onto and $ S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} $, then $ n(S) $ is equal to:

Let A be a 3 × 3 matrix such that $\text{det}(A) = 5$. If $\text{det}(3 \, \text{adj}(2A)) = 2^{\alpha \cdot 3^{\beta} \cdot 5^{\gamma}}$, then $ (\alpha + \beta + \gamma) $ is equal to:

If $ \alpha $ and $ \beta $ are the roots of the equation $ 2z^2 - 3z - 2i = 0 $, where $ i = \sqrt{-1} $, then \[ 16 \cdot {Re} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \cdot {Im} \left( \frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}} \right) \] is equal to:

Let $ \alpha, \beta $ be the roots of the equation $ x^2 - ax - b = 0 $ with $ \text{Im}(\alpha) < \text{Im}(\beta) $. Let $ P_n = \alpha^n - \beta^n $. If \[ P_3 = -5\sqrt{7}, \quad P_4 = -3\sqrt{7}, \quad P_5 = 11\sqrt{7}, \quad P_6 = 45\sqrt{7}, \] then $ |\alpha^4 + \beta^4| $ is equal to:

The sum of the squares of all the roots of the equation $ x^2 + [2x - 3] - 4 = 0 $ is:

For the thermal decomposition of $ N_2O_5(g) $ at constant volume, the following table can be formed, for the reaction mentioned below: \[ 2 N_2O_5(g) \rightarrow 2 N_2O_4(g) + O_2(g) \] Given: Rate constant for the reaction is $ 4.606 \times 10^{-2} \text{ s}^{-1} $.

O$_2$ gas will be evolved as a product of electrolysis of:
(A) an aqueous solution of AgNO3 using silver electrodes.
(B) an aqueous solution of AgNO3 using platinum electrodes.
(C) a dilute solution of H2SO4 using platinum electrodes.
(D) a high concentration solution of H2SO4 using platinum electrodes.
Choose the correct answer from the options given below :

The number of complex numbers $ z $, satisfying $ |z| = 1 $ and \[ \left| \frac{z}{\overline{z}} + \frac{\overline{z}}{z} \right| = 1, \] is:

If $ \alpha + i\beta $ and $ \gamma + i\delta $ are the roots of the equation $ x^2 - (3-2i)x - (2i-2) = 0 $, $ i = \sqrt{-1} $, then $ \alpha\gamma + \beta\delta $ is equal to:

If \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^2 \cos^2 x}{1 + e^x} dx = \pi(a\pi^2 + \beta), \quad a, \beta \in \mathbb{Z}, \] then $ (a + \beta)^2 $ equals:

What is the shape of the $ \text{ClO}_2^- $ ion according to VSEPR theory?

Let the mean and variance of the observations $ 2, 3, 3, 4, 5, 7, a, b $ be 4 and 2, respectively. Then, the mean deviation about the mode of the observation is:

Resonance in X$_2$Y can be represented as

The enthalpy of formation of X$_2$Y is 80 kJ mol$^{-1}$, and the magnitude of resonance energy of X$_2$Y is:

The element that does not belong to the same period of the remaining elements (modern periodic table) is:

If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \] has infinitely many solutions, then $ \lambda + \mu $ is equal to:}

An electric dipole of mass $ m $, charge $ q $, and length $ l $ is placed in a uniform electric field $ E = E_0 \hat{i} $. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be:

The equilibrium constant for decomposition of $ H_2O $ (g) $ H_2O(g) \rightleftharpoons H_2(g) + \frac{1}{2} O_2(g) \quad (\Delta G^\circ = 92.34 \, \text{kJ mol}^{-1}) $ is $ 8.0 \times 10^{-3} $ at 2300 K and total pressure at equilibrium is 1 bar. Under this condition, the degree of dissociation ($ \alpha $) of water is _____ $\times 10^{-2}$ (nearest integer value). [Assume $ \alpha $ is negligible with respect to 1]

Two iron solid discs of negligible thickness have radii $ R_1 $ and $ R_2 $ and moment of inertia $ I_1 $ and $ I_2 $, respectively. For $ R_2 = 2R_1 $, the ratio of $ I_1 $ and $ I_2 $ would be $ \frac{1}{x} $, where $ x $ is:

Statement-I: Melting point of neopentane is greater than that of n-pentane.
Statement-II: Neopentane gives only one mono-substituted product.