List of practice Questions

A dielectric sphere of radius \( R \) has constant polarization \( \mathbf{P} = P_0 \hat{z} \) so that the field inside the sphere is \( \mathbf{E} = - \frac{P_0}{3 \epsilon_0} \hat{z} \). Then, which of the following is (are) correct?
Consider a circular parallel plate capacitor of radius \( R \) with separation \( d \) between the plates (\( d \ll R \)). The plates are placed symmetrically about the origin. If a sinusoidal voltage \( V = V_0 \sin(\omega t) \) is applied between the plates, which of the following statement(s) is (are) true?
The linear mass density of a rod of length \( L \) varies from one end to the other as \( \lambda_0 \left( 1 + \frac{x^2}{L^2} \right) \), where \( x \) is the distance from one end with tensions \( T_1 \) and \( T_2 \) in them (see figure), and \( \lambda_0 \) is a constant. The rod is suspended from a ceiling by two massless strings. Then, which of the following statement(s) is (are) correct?
Unpolarized light is incident on a combination of a polarizer, a \( \lambda/2 \) plate and a \( \lambda/4 \) plate kept one after the other. What will be the output polarization for the following configurations?
Configuration 1: Axes of the polarizer, the \( \lambda/2 \) plate and the \( \lambda/4 \) plate are all parallel to each other.
Configuration 2: The \( \lambda/2 \) plate is rotated by 45° with respect to configuration 1.
Configuration 3: The \( \lambda/4 \) plate is rotated by 45° with respect to configuration 1.
A photon of frequency \( \nu \) strikes an electron of mass \( m \) initially at rest. After scattering at an angle \( \phi \), the photon loses half of its energy. If the electron recoils at an angle \( \theta \), which of the following is (are) true?
KC1 has the NaCl type structure which is fcc with two-atom basis, one at \( (0, 0, 0) \) and the other at \( \left( \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right) \). Assume that the atomic form factors of \( K^+ \) and \( Cl^- \) are identical. In an x-ray diffraction experiment on KC1, which of the following \( (h k l) \) peaks will be observed?
In the radiation emitted by a black body, the ratio of the spectral densities at frequencies \( 2\nu \) and \( \nu \) will vary with \( \nu \) as:
Consider Rydberg (hydrogen-like) atoms in a highly excited state with \( n \) around 300. The wavelength of radiation coming out of these atoms for transitions to the adjacent states lies in the range:
A white dwarf star has volume \( V \) and contains \( N \) electrons so that the density of electrons is \( n = \frac{N}{V} \). Taking the temperature of the star to be 0 K, the average energy per electron in the star is \( \epsilon_0 = \frac{3h^2}{10m} (3n)^{2/3 \), where \( m \) is the mass of the electron. The electronic pressure in the star is:}
Which of the following is due to inhomogeneous refractive index of earth’s atmosphere?
For the three matrices given below, which one of the choices is correct?
A plane in a cubic lattice makes intercepts of \( a, a/2 \) and \( 2a/3 \) with the three crystallographic axes, respectively. The Miller indices for this plane are:
Consider a system of \( N \) particles obeying classical statistics, each of which can have an energy 0 or \( E \). The system is in thermal contact with a reservoir maintained at a temperature \( T \). Let \( k \) denote the Boltzmann constant. Which one of the following statements regarding the total energy \( U \) and the heat capacity \( C \) of the system is correct?
If the Boolean function \( Z = PQ + PQR + PQRS + PQRST + PQRSTU \), then \( Z \) is:
The dispersion relation for electromagnetic waves travelling in a plasma is given as \( \omega^2 = c^2k^2 + \omega_p^2 \), where \( c \) and \( \omega_p \) are constants. In this plasma, the group velocity is:

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function 

Then \( E(X | Y = -1) \) equals} 
 

Let \( X_1, X_2, X_3 \) be independent random variables with the common probability density function 

Let \( Y = \min \{ X_1, X_2, X_3 \}, \, E(Y) = \mu_Y \, \text{and} \, \text{Var}(Y) = \sigma_Y^2. \) Then \( P(Y>\mu_Y + \sigma_Y) \) equals} 
 

Let \( X \) be a continuous random variable with the probability density function 

where \( k \) is a real constant. Then \( P(1<X<5) \) equals 
 

Let a unit vector \( \mathbf{v} = (v_1 \, v_2 \, v_3)^T \) be such that \( A\mathbf{v} = 0 \), where

\[ A = \begin{pmatrix} \frac{5}{6} & -\frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix}. \]

Then the value of \( \sqrt{6} (|v_1| + |v_2| + |v_3|) \) equals
Let

\[ u_n = \frac{18n + 3}{(3n - 1)^2 (3n + 2)^2}, \quad n \in \mathbb{N}. \]

Then

\[ \sum_{n=1}^{\infty} u_n \text{ equals} \]

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function 

Then \( 9 \, \text{Cov}(X, Y) \) equals 
 

If \( Y = \log_{10} X \) has \( N(\mu, \sigma^2) \) distribution with moment generating function \( M_Y(t) = e^{5t + 2t^2} \), for \( t \in (-\infty, \infty) \), then \( P(X<1000) \) equals

Let \( X_1, X_2, \dots, X_n \) (where \( n \geq 2 \)) be a random sample from a distribution with the probability density function 

Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete? 
 

Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability mass function 

If \( \hat{X}(X_1, X_2, X_3) \) is an unbiased estimator of \( \theta \), which of the following CANNOT be attained as a value of the variance of \( \hat{X} \) at \( \theta = 1 \)? 
 

Let \( X_1, X_2 \) be a random sample from a distribution with the probability mass function 

Which of the following is (are) unbiased estimator(s) of \( \theta \)?