List of practice Questions

Identify the correct statement(s) regarding nuclei:

In the Fourier series expansion of two functions \( f_1(t) = 4t^2 + 3 \) and \( f_2(t) = 6t^3 + 7t \) in the interval \( -\frac{T}{2} \) to \( +\frac{T}{2} \), the Fourier coefficients \( a_n \) and \( b_n \) (\( a_n \) and \( b_n \) are coefficients of \( \cos(n\omega t) \) and \( \sin(n\omega t) \), respectively) satisfy:

The radial component of acceleration in plane polar coordinates is given by:

A gaseous system, enclosed in an adiabatic container, is in equilibrium at pressure \( P_1 \) and volume \( V_1 \). Work is done on the system in a quasi-static manner due to which the pressure and volume change to \( P_2 \) and \( V_2 \), respectively, in the final equilibrium state. At every instant, the pressure and volume obey the condition \( PV^\gamma = C \), where \( \gamma = \frac{C_p}{C_v} \) and \( C \) is a constant. If the work done is zero, then identify the correct statement(s).

An isolated ideal gas is kept at a pressure \( P_1 \) and volume \( V_1 \). The gas undergoes free expansion and attains a pressure \( P_2 \) and volume \( V_2 \). Identify the correct statement(s).
\(\left(\gamma = \frac{C_p}{C_v}\right)\)

The rms velocity of molecules of oxygen gas is given by \( v = \sqrt{\frac{3kT}{m}} \) at some temperature \( T \). The molecules of another gas have the same rms velocity at temperature \( T' = \frac{T}{16} \). The second gas is:

A system undergoes a thermodynamic transformation from state \( S_1 \) to \( S_2 \) via two different paths 1 and 2. The heat absorbed and work done along path 1 are 50 J and 30 J, respectively. If the heat absorbed along path 2 is 30 J, the work done along path 2 is:

A linearly polarized light falls on a quarter-wave plate and the emerging light is found to be elliptically polarized. The angle between the fast axis of the quarter-wave plate and the plane of polarization of the incident light can be:

The expression for the magnetic field that induces the electric field \( \vec{E} = K(y\hat{i} + 3z\hat{j} + 4y\hat{k}) \cos(\omega t) \) is:

A particle, initially at the origin in an inertial frame \( S \), has a constant velocity \( V\hat{i} \). Frame \( S' \) is rotating about the \( z \)-axis with angular velocity \( \omega \) (anticlockwise). The coordinate axes of \( S' \) coincide with those of \( S \) at \( t = 0 \). The velocity of the particle \((V'_x, V'_y)\) in the \( S' \) frame, at \( t = \frac{\pi}{2\omega} \), is:

At \( t = 0 \), \( N_0 \) number of radioactive nuclei \( A \) start decaying into \( B \) with a decay constant \( \lambda_a \). The daughter nuclei \( B \) decay into nuclei \( C \) with a decay constant \( \lambda_b \). Then, the number of nuclei \( B \) at small time \( t \) (to the leading order) is:

Arrange the following telescopes, where \( D \) is the telescope diameter and \( \lambda \) is the wavelength, in order of decreasing resolving power:
I. \( D = 100\,\text{m}, \lambda = 21\,\text{cm} \)
II. \( D = 2\,\text{m}, \lambda = 500\,\text{nm} \)
III. \( D = 1\,\text{m}, \lambda = 100\,\text{nm} \)
IV. \( D = 2\,\text{m}, \lambda = 10\,\text{mm} \)

Metallic lithium has bcc crystal structure. Each unit cell is a cube of side \( a \). The number of atoms per unit volume is:

The moment of inertia of a solid sphere (radius \( R \) and mass \( M \)) about the axis which is at a distance of \( \frac{R}{2} \) from the center is:

Three events, \( E_1(ct = 0, x = 0) \), \( E_2(ct = 0, x = L) \) and \( E_3(ct = 0, x = -L) \) occur, as observed in an inertial frame \( S \). Frame \( S' \) is moving with a speed \( v \) along the positive x-direction with respect to \( S \). In \( S' \), let \( t'_1, t'_2, t'_3 \) be the respective times at which \( E_1, E_2, \) and \( E_3 \) occurred. Then,

The solution \( y(x) \) of the differential equation \( y\frac{dy}{dx} + 3x = 0 \), \( y(1) = 0 \), is described by:

Let \( M \) be a \( 2 \times 2 \) matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are:

A planet is in a highly eccentric orbit about a star. The distance of its closest approach is 300 times smaller than its farthest distance from the star. If the corresponding speeds are \( v_c \) and \( v_f \), then \( \frac{v_c}{v_f} \) is:

An object of density \( \rho \) is floating in a liquid with 75% of its volume submerged. The density of the liquid is:

An experiment with a Michelson interferometer is performed in vacuum using a laser of wavelength 610 nm. One beam passes through a glass cavity 1.3 cm long. When the cavity is filled with a medium of refractive index \( n \), 472 dark fringes move past a reference line. The speed of light is \( 3\times10^8 \) m/s. The value of \( n \) is:

The function \( e^{\cos x} \) is Taylor expanded about \( x = 0 \). The coefficient of \( x^2 \) is:

Consider the problem of testing \( H_0: X \sim f_0 \) against \( H_1: X \sim f_1 \) based on a sample of size 1, where \[ f_0(x) = \begin{cases} 1, & 0 \le x \le 1, \\ 0, & \text{otherwise,} \end{cases} \quad f_1(x) = \begin{cases} 2x, & 0 \le x \le 1, \\ 0, & \text{otherwise.} \end{cases} \] Then, the probability of Type II error of the most powerful test of size \(\alpha = 0.1\) is equal to

Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be independent and identically distributed random variables with probability density function \[ f(x) = \begin{cases} \dfrac{1}{x^2}, & x \ge 1, \\ 0, & \text{otherwise.} \end{cases} \] Then, which of the following random variables has/have finite expectation?

Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be a random sample from a distribution with probability density function \[ f(x; \theta) = \begin{cases} \dfrac{1}{2\theta}, & -\theta \le x \le \theta, \\ 0, & |x| > \theta, \end{cases} \] where \( \theta \in (0, \infty) \) is unknown. If \( R = \min\{X_1, X_2, \dots, X_n\} \) and \( S = \max\{X_1, X_2, \dots, X_n\} \), then which of the following statements is/are TRUE?

Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be a random sample from a distribution with probability density function \[ f(x; \theta) = \begin{cases} \dfrac{3x^2}{\theta} e^{-x^3 / \theta}, & x > 0, \\ 0, & \text{otherwise,} \end{cases} \] where \( \theta \in (0, \infty) \) is unknown. If \( T = \sum_{i=1}^n X_i^3 \), then which of the following statements is/are TRUE?