List of practice Questions

Which of the following option(s) is/are correct for photons?
Which of the following option(s) is/are correct for the ground state of a hydrogen atom?
A paramagnetic material containing paramagnetic ions with total angular momentum \( J = \frac{1}{2} \) is kept at absolute temperature \( T \). The ratio of the magnetic field required for 80% of the ions to be in the lowest energy state to that required for having 60% of the ions to be in the lowest energy state at the same temperature is:
Consider one mole of a monovalent metal at absolute zero temperature, obeying the free electron model. Its Fermi energy is \( E_F \). The energy corresponding to the filling of \( \frac{N_A}{2} \) electrons, where \( N_A \) is the Avogadro number, is \( 2^n E_F \). The value of \( n \) is:
Consider the function \( f(z) = \frac{1}{z^2 (z - 2)^3} \) of a complex variable \( z \).
The residues of the function at \( z = 0 \) and \( z = 2 \), respectively, are:
The nuclear energy levels of mirror nuclei are similar. Using this empirical fact alone, the nuclear force can be said to be independent of which one of the following properties of the nucleons?

A logic gate circuit is shown in the figure below. The correct combination for the input \( (P, Q) \) for which the output \( T = 1 \) is:


 

Which one of the following baryons has strangeness quantum number \( S = -1 \)?
As per the Drude model of metals, the electrical resistance of a metallic wire of length \( L \) and cross-sectional area \( A \) is:
(Consider \( \tau \) as the relaxation time, \( m \) as electron mass, \( n \) as carrier concentration, and \( e \) as electronic charge)
Which one of the following is correct for the phase velocity \( v_p \) and group velocity \( v_g \)? (c is the speed of light in vacuum)
The Joule-Thomson expansion of a gas is:
A particle of mass \( m \) is in a potential \( V(x) = \frac{1}{2} m \omega^2 x^2 \) for \( x > 0 \), and \( V(x) = \infty \) for \( x \leq 0 \), where \( \omega \) is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is:
Consider a crystal that has a basis of one atom. Its primitive vectors are \( \vec{a}_1 = a\hat{i} \), \( \vec{a}_2 = a\hat{j} \), \( \vec{a}_3 = \frac{a}{2}(\hat{i} + \hat{j} + \hat{k}) \), where \( \hat{i}, \hat{j}, \hat{k} \) are the unit vectors in the \( x \), \( y \), and \( z \) directions of the Cartesian coordinate system and \( a \) is a positive constant. Which one of the following is the correct option regarding the type of the Bravais lattice?
For a two-dimensional hexagonal lattice with lattice constant \( a \), the atomic density is:

An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of \( \frac{2\pi}{n} \), is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen? 
 

A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K). Which one of the following options displays the color codes that are consistent with the color model? 


 

Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?

“Why do they pull down and do away with crooked streets, I wonder, which are my delight, and hurt no man living? Every day the wealthier nations are pulling down one or another in their capitals and their great towns: they do not know why they do it; neither do I. It ought to be enough, surely, to drive the great broad ways which commerce needs and which are the life-channels of a modern city, without destroying all history and all the humanity in between: the islands of the past.”
(From Hilaire Belloc’s “The Crooked Streets”)
Based only on the information provided in the above passage, which one of the following statements is true?

The figures I, II, and III are parts of a sequence. Which one of the following options comes next in the sequence at IV?


 

A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting, and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?

As the police officer was found guilty of embezzlement, he was _________ dismissed from the service in accordance with the Service Rules. Select the most appropriate option to complete the above sentence.

Is there any good show _________ television tonight? Select the most appropriate option to complete the above sentence.

Consider the set \( \{ 1, x, x^2 \} \). An orthonormal basis in \( x \in [-1, 1] \) is formed from these three terms, where the normalization of a function \( f(x) \) is defined via \[ \int_{-1}^{1} \left[ f(x) \right]^2 \, dx = 1. \] If the orthonormal basis set is \( \left( \frac{\sqrt{3}}{2}, \frac{\sqrt{5}}{2} x, \frac{1}{2} \sqrt{\frac{21}{N}} (5x^2 - 3) \right) \), then the value of \( N \) (in integer) is:
In a one-dimensional Hamiltonian system with position \( q \) and momentum \( p \), consider the canonical transformation \( (q, p) \to (Q = \frac{1}{p}, P = q p^2) \), where \( Q \) and \( P \) are the new position and momentum, respectively. Which of the following option(s) regarding the generating function \( F \) is/are correct?
A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the \( x \)-axis. The rod itself is moving vertically upward along the \( z \)-direction against gravity with a constant speed, starting from \( z = 0 \) at \( t = 0 \), and remains horizontal. The conjugate momenta are denoted by \( p_x, p_y, p_z \) and the Hamiltonian by \( H \). Which of the following option(s) is/are correct?