List of practice Questions

If \( P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix} \) is the adjoint of a matrix \( A \) and \( \det(A) = 4 \), then the value of \( \alpha \) is:

If \( 11^{12} - 11^2 = k(5 \times 10^9 + 6 \times 10^9 + 33 \times 10^8 + 110 \times 10^7 + \ldots + 33) \), then find the value of \( k \).

If \( f(x) = (x+1)^2 - 1, x \ge -1 \), then \( \{x \mid f(x) = f^{-1}(x)\} \) is:

Let \( [t] \) denote the greatest integer function and \( [t - m] = [t] - m \) when \( m \in \mathbb{Z} \). If \( k = 2[2x - 1] - 1 \) and \( 3[2x - 2] + 1 = 2[2x - 1] - 1 \), then the range of \( f(x) = [k + 5x] \) is:

In a Carnot engine, if the absolute temperature of the source is \( 25% \) more than the absolute temperature of the sink, then the efficiency of the engine is

A force \( F \) is applied on a body of mass \( m \) so that the body starts moving from rest. The power delivered by the force at time \( t \) is proportional to:

Evaluate \( \int \frac{\sec^2 x}{\sin^7 x} \, dx - \int \frac{7}{\sin^7 x} \, dx \):

In triangle $ ABC $, if $ a = 13 $, $ b = 8 $, $ c = 7 $, then $ \cos(B+C) = $

In a triangle ABC, if \((r_1 - r_3)(r_1 - r_2) - 2r_2r_3 = 0\), then \(a^2 - b^2 =\)

If $ \alpha, \beta, \gamma $ are the roots of the equation $ x^3 + px^2 + qx + r = 0 $, then $ \alpha^3 + \beta^3 + \gamma^3 = $

In an LCR series circuit, if the potential differences across inductor, capacitor, and resistor are $60~\text{V}$, $30~\text{V}$, and $40~\text{V}$ respectively, then the net voltage applied to the circuit is

When a sphere is taken to the bottom of a sea of depth $1~\text{km}$, it contracts in volume by $0.01%$. Then the bulk modulus of the material of the sphere is (Acceleration due to gravity = $10~\text{m/s}^2$)

The increasing order of number of lone pair electrons on the central atom of the following molecules is which of the following?
I) ${ClF3}$
II) ${XeF2}$
III) ${SF4}$
IV) ${SiH4}$

The number of primary (1$^\circ$), secondary (2$^\circ$), and tertiary (3$^\circ$) alcohols possible for the formula C$_5$H$_{12}$O respectively are:

Match the following:

List-I (Atomic number, Z)	List-II (Block)
A) 112	I) \(s\)
B) 116	II) \(p\)
C) 88	III) \(d\)
D) 100	IV) \(f\)

Observe the following reactions
I) Sucrose (aq) + H2O →[x] glucose + fructose
II) Glucose (aq) →[y] ethanol + CO2
What are $x$ and $y$ respectively? Identify the correct option from the following:

The set of all real values of $x$ such that \[ f(x) = \frac{[x] - 1}{\sqrt{[x]^2 - [x] - 6}} \] is a real valued function is

If a function $f : \mathbb{Z} \to \mathbb{Z}$ is defined by $f(x) = x - (-1)^x$, then $f(x)$ is

If $(3 + 4i)^{2025} = 5^{2023}(x + iy)$, then find $\sqrt{x^2 + y^2}$.

If \[ \left(\frac{\cos \theta + i \sin \theta}{\sin \theta + i \cos \theta}\right)^{2024} + \left(\frac{1 + \cos \theta + i \sin \theta}{1 - \cos \theta + i \sin \theta}\right)^{2025} = x + iy, \] and $x + y$ at $\theta = \frac{\pi}{2}$ is

If \(x\) is a real number, then the number of solutions of \(\tan^{-1}\left(\sqrt{x(x+1)}\right) + \sin^{-1}\left(\sqrt{x^2 + x + 1}\right) = \dfrac{\pi}{2}\) is

Domain of the real-valued function \(f(x) = \log(x^2 - 1) + x \, \coth^{-1}x\) is

One of the latus recta of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ subtends an angle $2 \tan^{-1} \left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2 = 36$ and $e$ is the eccentricity of the hyperbola, then find $\sqrt{a^2 + e^2}$.

If $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$, then the foot of the perpendicular drawn from the focus of the parabola onto the normal $L$ is:

If the acute angle between the circles $S \equiv x^2 + y^2 + 2kx + 4y - 3 = 0$ and $S^1 \equiv x^2 + y^2 - 4x + 2ky + 9 = 0$ is $\cos^{-1}\left(\frac{3}{8}\right)$ and the centre of $S^1 = 0$ lies in the first quadrant, then the radical axis of $S = 0$ and $S^1 = 0$ is: