List of practice Questions

The magnitude of an electric field which can just suspend a deuteron of mass \( 3.2 \times 10^{-27} \) kg freely in air is:
Two charges \(5\) nC and \(-2\) nC are placed at points \( (5,0,0) \) and \( (23,0,0) \) in a region of space where there is no other external field. The electrostatic potential energy of this charge system is:
The space between the plates of a parallel plate capacitor is halved and a dielectric medium of relative permittivity \( 10 \) is introduced between the plates. The ratio of the final and initial capacitances of the capacitor is:
The efficiency of a Carnot heat engine is 25% and the temperature of its source is 127°C. Without changing the temperature of the source, if the absolute temperature of the sink is decreased by 10%, the efficiency of the engine is:
The total internal energy of 2 moles of a monoatomic gas at a temperature 27°C is \( U \). The total internal energy of 3 moles of a diatomic gas at a temperature 127°C is:
The fundamental frequency of an open pipe is 100 Hz. If the bottom end of the pipe is closed and \( \frac{1}{3} \) of the pipe is filled with water, then the fundamental frequency of the pipe is:
Two cylindrical vessels A and B of different areas of cross-section kept on the same horizontal plane are filled with water to the same height. If the volume of water in vessel A is 3 times the volume of water in vessel B, then the ratio of the pressures at the bottom of the vessels A and B is:
The acceleration of a body sliding down the inclined plane, having coefficient of friction \( \mu \), is
A body of 2 kg mass slides down with an acceleration of \( 4 \, \text{ms}^{-2} \) on an inclined plane having a slope of \( 30^\circ \). The external force required to take the same body up the plane with the same acceleration will be (Acceleration due to gravity \( g = 10 \, \text{ms}^{-2} \))
A body of mass \( 30 \) kg moving with a velocity \( 20 \) ms\(^{-1}\) undergoes one-dimensional elastic collision with another ball of the same mass moving in the opposite direction with a velocity of \( 30 \) ms\(^{-1}\). After collision, the velocities of the first and second bodies respectively are:
A force of \( (4\hat{i} + 2\hat{j} + \hat{k}) \) N is acting on a particle of mass \( 2 \) kg displaces the particle from a position of \( (2\hat{i} + 2\hat{j} + \hat{k}) \) m to a position of \( (4\hat{i} + 3\hat{j} + 2\hat{k}) \) m. The work done by the force on the particle in joules is:
A ring and a disc of same mass and same diameter are rolling without slipping. Their linear velocities are same, then the ratio of their kinetic energy is:
In the equation \( \left( P + \frac{a}{V^2} \right) (V - b) = RT \), where \( P \) is pressure, \( V \) is volume, \( T \) is temperature, \( R \) is the universal gas constant, and \( a, b \) are constants. The dimensions of \( a \) are:
Evaluate the integral: \[ \int_{-2}^{2} (4 - x^2)^{\frac{5}{2}} \, dx. \]
Evaluate the following limit: \[ \lim_{x \to \infty} \left[ \left(1 + \frac{1}{n^3} \right)^{\frac{1}{n^3}} \left(1 + \frac{8}{n^3} \right)^{\frac{8}{n^3}} \left(1 + \frac{27}{n^3} \right)^{\frac{9}{n^3}} \dots (2n)^{\frac{1}{n}} \right]. \]
Evaluate the integral: \[ \int e^{4x^2 + 8x -4} (x+1) \cos(3x^2 + 6x -4) \, dx.= \]
Evaluate the integral: \[ \int \left[ (\log_2 x)^2 + 2 \log_2 x \right] dx. \]
Evaluate the integral: \[ \int \frac{1}{(1 + x^2) \sqrt{x^2 + 2}} \, dx. \]
Evaluate the integral: \[ \int \sin^4 x \cos^4 x \, dx. \]
Evaluate the integral: \[ \int_{0}^{1} \sqrt{\frac{2 + x}{2 - x}} \, dx \]
The semi-vertical angle of a right circular cone is \( 45^\circ \). If the radius of the base of the cone is measured as 14 cm with an error of \( \left(\frac{\sqrt{2}-1}{11} \right) \) cm, then the approximate error in measuring its total surface area is (in sq. cm).
If a man of height 1.8 m is walking away from the foot of a light pole of height 6 m with a speed of 7 km per hour on a straight horizontal road opposite to the pole, then the rate of change of the length of his shadow is (in kmph):
If \[ f(x) = 5 \cos^3 x - 3 \sin^3 x \quad \text{and} \quad g(x) = 4 \sin^3 x + \cos^2 x, \] then the derivative of \( f(x) \) with respect to \( g(x) \) is:
Evaluate the limit: \[ \lim\limits_{x \to \infty} \frac{[2x - 3]}{x}. \]
Evaluate the limit: \[ \lim\limits_{x \to 0} \frac{\cos 2x - \cos 3x}{\cos 4x - \cos 5x}.= \]