List of practice Questions

A carnot engine works between the temperatures $327^\circ\text{C}$ and $227^\circ\text{C}$. If the work output of the engine is $1\text{ kJ}$, then the amount of heat absorbed by the engine is

$\gamma_1$ is the ratio of the specific heat capacities of the rigid diatomic gas at constant pressure to that at constant volume and $\gamma_2$ is the corresponding value for a non-rigid diatomic gas molecule with an additional vibrational mode, then the ratio of $\gamma_1$ to $\gamma_2$ is

Bernoulli’s equation holds good for

Two copper wires have the lengths in the ratio $1 : 2$ and their radii are in the ratio $3 : 1$. If they are stretched by the same force, the ratio of the respective longitudinal strains in the two wires is

The escape velocities of two planets $A$ and $B$ are in the ratio $2 : 3$. If the ratio of their radii is $3 : 4$, then the ratio of acceleration due to gravity at the surface of the planet $A$ to that at the surface of the planet $B$ is

The height above the surface of the earth at which the acceleration due to gravity becomes $\frac{g}{9}$ in terms of radius of earth $R$ is ($g$ is acceleration due to gravity at the surface of the earth)}

The torque transmitted by an engine to a rotor in $100 \text{ Nm}$. If the power of the engine is $15 \text{ kW}$, the angular speed of the engine is

If the moment of inertia, rotational kinetic energy and angular momentum of a body are $I$, $E$ and $L$ respectively, then,

In an elastic collision,

The kinetic energy of a body of mass $5 \text{ kg}$ having a linear momentum $4 \text{ kg ms}^{-1}$ is

What will be the maximum speed of a car on a circular road of radius $12 \text{ m}$ if the coefficient of friction between the tyres and the road is $0.3$? $g = 10 \text{ms}^{-2}$}

The linear momentum of a particle as a function of time is given as $p = (3t^2 + 2t + 1)$ kgms$^{-1}$. Then, the force acting on the particle at $t = 3s$ will be}

A body is projected up with a velocity of $30 \text{ ms}^{-1}$ at an angle of $30^\circ$. The ratio of maximum height reached to the height reached in the first second is ($g = 10 \text{ms}^{-2}$)}

A vector is given as $\vec{A} = 3\hat{j} - 4\hat{k}$. The vector parallel to $\vec{A}$ and magnitude the same as that of the vector $\hat{i} - 2\hat{j}$ is

A car travels half the distance with a velocity of $20 \text{ kmh}^{-1}$ and another half distance with a velocity of $30 \text{ kmh}^{-1}$ along a straight road. The average velocity of the car in $\text{km h}^{-1}$ is

The number of significant figures in $50000.040 \times 10^{-3}$ is

The dimensions of Planck's constant are the same as those of

The minimum of the following linear programming problem occurs at:
Minimize $C = 7x + 10y$
subject to $x + y \geq 3$, $x + 2y \geq 4$, $x, y \geq 0$

The integrating factor of the differential equation $2dy = (y + \cos x) dx$ is

The general solution of the differential equation $x^3 \frac{dy}{dx} + 3x^2 y = \cos x$ is

The value of the integral $\int_{0}^{\pi / 2} \sqrt{\cos x \sin 2x} dx$ is equal to

A curve with equation $y = x^3 - 8x^2 + 16x$ meets the $x$-axis at the origin $O$ and at a point $A$. Then the area of the region, bounded by the curve and the straight-line segment $OA$, is

Let $f(x) = \frac{1}{20}(x - 5)^2, x \in \mathbb{R}$. If $\int_{-5}^{5} f(x) dx = \int_5^{a} f(x) dx$, where $a > 5$ is a real constant, then the value of $a$ is equal to

The value of $\int_{-1}^{1} \frac{\log_e(1 + |x|)}{1 + |x|} dx$ is equal to

$\int \frac{\sqrt{\sqrt{x} + 1}}{\sqrt{x}} dx = $}