List of practice Questions

The function $ f: (-\infty, \infty) \to (-\infty, 1) $, defined by \[ f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}, \] is:

Let $ \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} $, $ \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) $ and $ \mathbf{c} = \mathbf{b} \times \hat{k} $. Then the projection of $ \mathbf{c} - 2\hat{j} $ on $ \mathbf{a} $ is:

The equation of the chord of the ellipse $ \frac{x^2}{25} + \frac{y^2}{16} = 1 $, whose mid-point is $ (3, 1) $ is:

In an arithmetic progression, if $ S_{40} = 1030 $ and $ S_{12} = 57 $, then $ S_{30} - S_{10} $ is equal to:

Let $ A = [a_{ij}] $ be a square matrix of order 2 with entries either 0 or 1. Let $ E $ be the event that $ A $ is an invertible matrix. Then the probability $ P(E) $ is:

Let $A$ be the area of the region
$\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}$.
Then $540 A$ is equal to ______.

Acceleration due to gravity on the surface of earth is $ g $. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is ________________ g.

In a binary search tree with the following elements: $ 10, -4, 15, 13, 20, 5, 16, 19 $, the number of edges from node 19 to the root is?

The sum of all rational terms in the expansion of $ \left( 1 + 2^{1/3} + 3^{1/2} \right)^6 $ is equal to

Standard entropies of $X_2$, $Y_2$ and $XY_5$ are 70, 50, and 110 J $K^{-1}$ mol$^{-1}$ respectively. The temperature in Kelvin at which the reaction \[ \frac{1}{2} X_2 + \frac{5}{2} Y_2 \rightarrow XY_5 \quad \Delta H = -35 \, {kJ mol}^{-1} \] will be at equilibrium is (nearest integer):

Given below are two statements:
Statement I: The conversion proceeds well in a less polar medium. \[ {CH}_3{CH}_2{CH}_2{CH}_2{Cl} \xrightarrow{{HO}^-} {CH}_3{CH}_2{CH}_2{CH}_2{OH} + {Cl}^- \] Statement II: The conversion proceeds well in a more polar medium. \[ {CH}_3{CH}_2{CH}_2{CH}_2{Cl} \xrightarrow{{R}_3{N}} {CH}_3{CH}_2{CH}_2{CH}_2{NH}_2 + {Cl}^- \] In the light of the above statements, choose the correct answer from the options given below:

For positive integers $ n $, if $ 4 a_n = \frac{n^2 + 5n + 6}{4} $ and \[ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} \]

Let A, B, C be three points in the xy-plane, whose position vectors are given by $ \sqrt{3} \hat{i} + \hat{j} $, $ \hat{i} + \sqrt{3} \hat{j} $, and $ a\hat{i} + (1-a) \hat{j }$ respectively with respect to the origin $ O $. If the distance of the point C from the line bisecting the angle between the vectors $ \overrightarrow{OA} $ and $ \overrightarrow{OB} $ is $ \frac{9}{\sqrt{2}} $, then the sum of all possible values of $ a $ is:

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Knowing initial position $ x_0 $, and initial momentum $ p_0 $ is enough to determine the position and momentum at any time $ t $ for a simple harmonic motion with a given angular frequency $ \omega $.
Reason (R): The amplitude and phase can be expressed in terms of $ x_0 $ and $ p_0 $.
In the light of the above statements, choose the correct answer from the options given below:

A parallel plate capacitor of capacitance 1 μF is charged to a potential difference of 20 V. The distance between plates is 1 μm. The energy density between the plates of the capacitor is:

The frequency of revolution of the electron in Bohr’s orbit varies with $ n $, the principal quantum number as:

A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at the 40 cm mark. A mass of 400 g is suspended at the 10 cm mark. To maintain the balance of the rod, the mass to be suspended at the 90 cm mark is:

A uniform magnetic field of $ 0.4 $ T acts perpendicular to a circular copper disc $ 20 $ cm in radius. The disc is having a uniform angular velocity of $ 10\pi $ rad/s about an axis through its center and perpendicular to the disc. What is the potential difference developed between the axis of the disc and the rim? ($\pi = 3.14$)

Which of the following phenomena cannot be explained by the wave theory of light?

A 400 g solid cube having an edge of length $10$ cm floats in water. How much volume of the cube is outside the water? (Given: density of water = $1000 { kg/m}^3$)

Match List - I with List - II.