List of practice Questions

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) 112I) \(s\)
B) 116II) \(p\)
C) 88III) \(d\)
D) 100IV) \(f\)
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:
If \((1+x)^n = \sum_{r=0}^n \binom{n}{r} x^r\), then the value of \[ C_0 + (C_0 + C_1) + (C_0 + C_1 + C_2) + \cdots + (C_0 + C_1 + \cdots + C_n) \] is:
Consider the following statements:
Statement-I: CH$_3$NH$_2$ is more basic than NH$_3$, but C$_6$H$_5$NH$_2$ is less basic than NH$_3$.
Statement-II: The order of basic strength of amines in aqueous phase follows
(C$_2$H$_5$)$_2$NH > C$_2$H$_5$NH$_2$ > C$_6$H$_5$NH$_2$
The structure of the nitrogen-containing heterocyclic base shown below represents:
The most acidic carboxylic acid is: