List of practice Questions

A compass needle oscillates 20 times per minute at a location with dip \( 45^\circ \) and magnetic field \( B_1 \). It oscillates 30 times per minute at another location where dip is \( 30^\circ \) and field is \( B_2 \). Find the ratio \( \frac{B_1}{B_2} \).
A Carnot engine operates between a source and sink. The efficiency of the engine is 40\% and the sink temperature is \( 27^\circ \text{C} \). If the efficiency is to be increased to 50\%, then by how much should the source temperature be increased?
A car engine has a power of 20 kW and runs for 1 hour. If the thermal efficiency is 40\% and ambient temperature is 300 K, then what is the energy generated by fuel combustion?
The number of vibrational degrees of freedom for a diatomic molecule is:
A body is suspended from a string of length 1 m and mass 2 g. The mass of the body required to produce a fundamental frequency of 100 Hz in the string is (Take \( g = 10 \, \text{ms}^{-2} \))
Electrostatic force between two identical charges placed at a distance \( r \) is \( F \). A slab of width \( \frac{r}{5} \) and dielectric constant 9 is inserted between them. Then, the new force between the charges is:
A ray is incident from a medium of refractive index 2 to another with refractive index 1. The critical angle is:
An electric dipole with dipole moment \( 5 \times 10^{-7} \, \text{Cm} \) is in an electric field of \( 2 \times 10^4 \, \text{NC}^{-1} \) at an angle of \( 60^\circ \) with the direction of the field. The torque acting on the dipole is:
A hydrometer executes SHM when pushed into a liquid of density \( \rho \). If the mass of hydrometer is \( m \) and the radius of the tube is \( r \), then the time period of oscillation is:
An object undergoing SHM takes 0.5 s to travel from one point of zero velocity to the next such point. The angular frequency \( \omega \) is:
Same tension is applied to four wires made of the same material. The elongation is maximum in which wire?
A cone with half the density of water is floating in water as shown. It is depressed down by a small distance \( \delta \ll H \) and released. The frequency of SHM of the cone is:
A cone with half the density of water is floating in water
A sphere of surface area \( 4 \, \text{m}^2 \), temperature 400 K, emissivity 0.5 is in an environment of 200 K. The net rate of energy exchange is (take \( \sigma = 5.67 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \))
Two rectangular blocks of masses 40 kg and 60 kg are connected by a string on a frictionless table. A force of 1000 N is applied on the 60 kg block away from the 40 kg block. Find the tension in the string.
A ball of mass 0.5 kg moving horizontally at \( 10\, \text{ms}^{-1} \) strikes a vertical wall and rebounds with speed \( V \). The change in linear momentum is 8.0 kg·m/s. Find the magnitude of \( V \).
A mass of 1 kg falls from a height of 1 m and lands on a massless platform supported by a spring with spring constant \( 15 \, \text{Nm}^{-1} \). The maximum compression of the spring is (take \( g = 10 \, \text{ms}^{-2} \))
A mass of 1 kg falls from a height of 1 m
A bead of mass 400 g is moving along a straight line under a force delivering constant power 1.2 W. If the bead is initially at rest, the speed attained after 6 seconds is (in \( \text{ms}^{-1} \))
Masses \( m \left(\frac{1}{3}\right)^N \) are placed at \( x = N \), for \( N = 2, 3, 4, \dots \). If the total mass is \( M \), then the center of mass of the system is:
Consider a disc of radius \( R \) and mass \( M \). A hole of radius \( \frac{R}{3} \) is created in the disk such that the center of the hole is \( \frac{R}{3} \) away from the center of the disc. The moment of inertia of the remaining system about the axis perpendicular to the disc and passing through the original center is:
\( \lim_{n \to \infty} \left( \frac{1}{1 + n^5} + \frac{2^4}{2^5 + n^5} + \frac{3^4}{3^5 + n^5} + \ldots + \frac{n^4}{n^5 + n^5} \right) = \)
If the solution of \( \frac{dy}{dx} = y \log_e 0.5 = 0 \), \( y(0) = 1 \), and \( y(x) \to k \) as \( x \to \infty \), then \( k = \)
At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:
The function \( y = Ae^x + Be^{-2x} \) satisfies which of the following differential equations?
If \( N_A, N_B, N_C \) are the number of significant figures in \( A = 0.001204 \) m, \( B = 43120000 \) m and \( C = 1.200 \) m respectively, then which of the following is correct?
A car covers a distance at a speed of 60 km/h. It returns to the starting point at a speed of \( V \). If the average speed for the entire trip is 48 km/h, then what is the magnitude of \( V \)?