1.    The relative magnetic permeability of a type-I superconductor is

    (a) 0

    (b) –1

    (c)

    (d)

Ans.    (a)

Sol.    

    

2.    Considering baryon number and lapton number conservation laws, which of the following processes is/are allowed?

    

    (a) both (i) and (ii)

    (b) only (i)

    (c) only (ii)

    (d) neither (i) nor (ii)

Ans.    (c)

Sol.    (i)    

        B : +1 0 0 0 : Not conserved

        Therefore this is not all allowed process

    (ii)    

        Since neutrino is involve, therefore parity is violated. This is allowed through weak interactions.

3.    For the following circuit, what is the magnitude of V_out if V_in = 1.5 V?

    (a) 0.015 V

    (b) 0.15 V

    (c) 15 V

    (d) 150 V

Ans.    (c)

Sol.    

4.    For the differential equation , where n is a constant, the product of its two independent solutions is

    (a)

    (b) x

    (c) xⁿ

    (d)

Ans.    (b)

Sol.    This is a Euler-Cauchy by differential equation whose characteristic equation is

    m² – m – n(n + 1) = 0

    

    or m = 1 + 1, or m = –n

    Therefore two independent solution are y₁ = x^{1 + n} and y₂ = x^–n

    Therefore, y₁y₂ = x^{1 + n – n} = x

5.    Consider a one-dimensional gas of N non-interacting particles of mass m with the Hamiltonian for a single particle given by,

        

    The high temperature specific heat in units of R = Nk_B (k_B is the Boltzmann constant) is

    (a) 1

    (b) 1.5

    (c) 2

    (d) 2.5

Ans.    (c)

Sol.    

    

6.    An electric field is applied to a Hydrogen atom in n = 2 excited state. Ignoring spin, the n = 2 state is fourfold degenerate, which in the basis are given by and . If H' is the interaction Hamiltonian corresponding to the applied electric field, which of the following matrix elements is nonzero?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    The correct option is (c).

7.    A large number N of ideal bosons, each of mass m, are trapped in a three-dimensional potential . The bosonic system is kept at temperature T which is much lower than the Bose-Einstein condensation temperature T_c. The chemical potential (µ) satisfies

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    The correct option is (a).

8.    During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix , the unit vector along the axis of rotation is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    Since the vector along the axis of rotation remain unchanged during rotation then

    Ax = x    ...(i)

    

    Equation (i) is a standared eigenvalue-eigenvector relation for = 1. Equation (i) can be written

    or    x₂ = x₃    ...(ii)

    or    x₂ = – x₃    ...(iii)

    or    x₁ = –x₃    ...(iv)

    Using these relations we see that the general eigenvector is

    General eigenvector =

    Therefore a unit eigenvector along the axis of rotation is

    Unit eigenvector =

9.    For a spin particle, let and denote its spin up and spin down states, respectively. If and are composite states of two such particles, which of the following statements is true for their total spin S?

    (a) S = 1 for , and is not an eigenstate of the operator

    (b) is not an eigenstate of the operator , and S = 0 for

    (c) S = 0 for , and S = 1 for

    (d) S = 1 for , and S = 0 for

Ans.    (d)

Sol.    S = 1 is triplet and S = 0 for singlet for .

10.    Consider a transformation from one set of generalized coordinate and momentum (q, p) to another set (Q, P) denoted by,

        Q = pq^s;    P = q^r

    where s and r are constants. The transformation is canonical if

    (a) s = 0 and r = 1

    (b) s = 2 and r = –1

    (c) s = 0 and r = –1

    (d) s = 2 and r = 1

Ans.    (b)

Sol.    

11.    In order to estimate the specific heat of phonons, the appropriate method to apply would be

    (a) Einstein model for acoustic phonons and Debye model for optical phonons

    (b) Einstein model for optical phonons and Debye model for acoustic phonons

    (c) Einstein model for both optical and acoustic phonons

    (d) Debye model for both optical and acoustic phonons

Ans.    (b)

Sol.    At low temperature, the optical branch phonons have energies higher than k_BT and therefore, optical branch waves are not excited. And Debye model is not suitable for optical branch instead it is suitable for acoustical branch. Whereas Einstein model is useful for high temperature and therefore can be applied to optical branch.

12.    The pole of the function f(z) = cot z at z = 0 is

    (a) a removable singularity

    (b) an essential singularity

    (c) a simple pole

    (d) a second order pole

Ans.    (c)

Sol.    f(z) = cot z at z = 0

    

13.    A massive particle X in free space decays spontaneously into two photons. Which of the following statements is true for X?

    (a) X is charged

    (b) Spin of X must be greater than or equal to 2

    (c) X is a boson

    (d) X must be a baryon

Ans.    (c)

Sol.    

    The spin of X can be either 0, 1 or 2. (integer)

    Therefore, option (b) is wrong while option (c) is correct.

14.    The electric field of an electromagnetic wave is given by + . The wave is

    (a) linearly polarized at an angle from the x-axis

    (b) linearly polarized at an angle from the x-axis

    (c) elliptically polarized in clockwise direction when seen travelling towards the observer

    (d) elliptically polarized in counter-clockwise direction when seen travelling towards the observer

Ans.    (d)

Sol.    

    

    

15.    The nuclear spin and parity of in its ground state is

    (a) 0⁺

    (b) 0^–

    (c) 1⁺

    (d) 1^–

Ans.    (a)

Sol.     is an even-even nuclei, therefore I = 0, P = +ve

    Spin-parity = 0⁺

16.    An infinitely long thin cylindrical shell has its axis coinciding with the z-axis. It carries a surface charge density , where is the polar angle and is a constant. The magnitude of the electric field inside the cylinder is

    (a) 0

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

17.    Consider a three-dimensional crystal of N inert gas atoms. The total energy is given by , where p = 12.13, q = 14.45, and R is the nearest neighbour distance between two atoms. The two constants, and R, have the dimensions of energy and length, respectively. The equilibrium separation between two nearest neighbour atoms in units of σ (rounded off to two decimal places) is ____________

Ans.    1.09

Sol.    

    

    

18.    The energy-wavevector (E-k) dispersion relation for a particle in two dimensions is E = Ck, where C is a constant. If its density of states D(E) is proportional to E^p then the value of p is ______________

Ans.    1

Sol.    For E(k) k^s. The density of states in d-dimension is

    Given E = Ck    s = 1, d = 2

    

19.    A circular loop made of a thin wire has radius 2 cm and resistance 2 . It is placed perpendicular to a uniform magnetic field of magnitude Tesla. At time t = 0 the field starts decaying as , where t₀ = 1s. The total charge that passes through a cross section of the wire during the decay is Q. The value of Q in µC (rounded off to two decimal places) is __________

Ans.    6.28

Sol.    

    

20.    The electric field of an electromagnetic wave in vacuum is given by

        .

    The wave is reflected from the z = 0 surface. If the pressure exerted on the surface is , the value of (rounded off to one decimal place) is ______________

Ans.    0.8

Sol.    

    

21.    The Hamiltonian for a quantum harmonic oscillator of mass m in three dimensions is where is the angular frequency. The expectation value of r² in the first excited state of the oscillator in units of (rounded off to one decimal place) is _________

Ans.    2.5

Sol.    

    For first excited state n_x = 1, n_y = 0, n_z = 0

    Hence, it is triply degenerate one can take

    n_x = 0, n_y = 1, n_z = 0 or n_x = 0, n_y = 0, n_z = 1

    putting any one combination, expectation value of r² =

22.    The Hamiltonian for a particle of mass m is where q and p are the generalized coordinate and momentum, respectively, t is time and k is a constant. For the initial condition, q = 0 and p = 0 at t = 0, . The value of is ________________

Ans.    3

Sol.    

    

23.    At temperature T Kelvin (K), the value of the Fermi function at an energy 0.5 eV above the Fermi energy is 0.01. Then T, to the nearest integer, is _____________

    (k_B = 8.62 × 10^–5 eV/K)

Ans.    1262

Sol.    

    

24.    Let represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of and is given by a density matrix that is defined as . If c₁ = 0.4 and c₂ = 0.6, the matrix element (rounded off to one decimal place) is _________

Ans.    0.6

Sol.    

    c₂ = 0.6

25.    A conventional type-I superconductor has a critical temperature of 4.7 K at zero magnetic field and a critical magnetic field of 0.3 Tesla at 0 K. The critical field in Tesla at 2 K (rounded off to three decimal places) is _____________

Ans.    0.246

Sol.    

        = 0.3 [1 – 0.181] = 0.3 × 0.819 = 0.246 Atm

26.    Consider the following Boolean expression:

        

    It can be represented by a single three-input logic gate. Identify the gate.

    (a) AND

    (b) OR

    (c) XOR

    (d) NAND

Ans.    (d)

Sol.    

    

    

    

27.    The value of the integral , where k > 0 and a > 0, is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

28.    The wave function of a particle is as shown below

    Here K is a constant, and a > d. The position uncertainty of the particle is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

    

    Hence wavefunction is symmetric about x = 0, so

    

    

    

    

29.    A solid cylinder of radius R has total charge Q distributed uniformly over its volume. it is rotating about its axis with angular speed . The magnitude of the total magnetic moment of the cylinder is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

30.    Consider the motion of a particle along the x-axis in a potential V(x) = F|x|. Its ground state energy E₀ is estimated using the uncertainty principle. Then E₀ is proportional to

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    

    

    For minimum energy,

    

31.    A 3-bit analog-to-digital converter is designed to digitize analog signals ranging from 0 V to 10V. For this converter, the binary output corresponding to an input of 6 V is

    (a) 011

    (b) 101

    (c) 100

    (d) 010

Ans.    (c)

Sol.    

    

    

    

32.    The Hamiltonian operator for a two-level quantum system is . If the state of the system at t = 0 is given by then at a later time t is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

33.    A particle of mass m moves in a lattice along the x-axis in a periodic potential V(x) = V(x + d) with periodicity d. The corresponding Brillouin zone extends from –k₀ to k₀ with these two k-points being equivalent. If a weak force F in the x-direction is applied to the particle, it starts a periodic motion with time period T. Using the equation of motion for a particle moving in a band, where p_crystal is the crystal momentum of the particle, the period T is found to be (h is Planck constant)

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    

    Using Heisenberg uncertainty . Thus correct option is (d).

34.    Consider a potential barrier V(x) of the form:

    where V₀ is a constant. For particles of energy E < V₀ incident on this barrier from the left, which of the following schematic diagrams best represents the probability density as a function of x.

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    The correct option is (a).

35.    The spin-orbit interaction term of an electron moving in a central field is written as , where r is the radial distance of the electron from the origin. If an electron moves inside a uniformly charged sphere, then

    (a) f(r) = constant

    (b) f(r) r^–1

    (c) f(r) r^–2

    (d) f(r) r^–3

Ans.    (a)

Sol.    The electric potential of a uniformly charged sphere at r < R is

        

    where Q is the electric charge on the sphere of radius R and k is a constant.

    The interaction energy is W = , where for central potential V,

     = constant. Thus option (a) is correct.

36.    For the following circuit, the correct logic values for the entries X₂ and Y₂ in the truth table are

    

    (a) 1 and 0

    (b) 0 and 0

    (c) 0 and 1

    (d) 1 and 1

Ans.    (a)

Sol.    The correct option is (a).

37.    In a set of N successive polarizers, the m^th polarizer makes an angle with the vertical. A vertically polarized light beam of intensity I₀ is incident on two such sets with N = N₁ and N = N₂, where N₂ > N₁. Let the intensity of light beams coming out be I(N₁) and I(N₂), respectively. Which of the following statements is correct about the two outgoing beams?

    (a) I(N₂) > I(N₁); the polarization in each case is vertical

    (b) I(N₂) < I(N₁); the polarization in each case is vertical

    (c) I(N₂) > I(N₁); the polarization in each case is horizontal

    (d) I(N₂) < I(N₁); the polarization in each case is horizontal

Ans.    (c)

Sol.    

    I(N₂) > I(N₁)

    I(5) = I₀[cos(18^*)]¹⁰ = 0.605 I₀

    N₂ = 10

    I(10) = I₀[cos(9^*)]²⁰ = 0.780 I₀

    I(10) > I(5)

38.    A ball bouncing off a rigid floor is described by the potential energy function

        V(x) = mgx for x > 0

         = for x < 0

    Which of the following schematic diagrams best represents the phase space plot of the ball?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.     p² + 2m (E – mgx) which is equation of parabola.

39.    An infinitely long wire parallel to the x-axis is kept at z = d and carries a current I in the positive x direction above a superconductor filling the region z < 0 (see figure). The magnetic field inside the superconductor is zero so that the field just outside the superconductor is parallel to its surface. The magnetic field due to this configuration at a point (x, y, z > 0) is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    Verify that = 0, when d = 0.

40.    The vector potential inside a long solenoid, with n turns per unit length and carrying current I, written in cylindrical coordinates is . If the term , where , is added to , the magnetic field remains the same if

    (a)

    (b)

    (c)

    (d)

    

Ans.    (d)

Sol.    

    

    

41.    Low energy collision (s-wave scattering) of pion with deuteron (d) results in the production of two protons . The relative orbital angular momentum (in units of ) of the resulting two-proton system for this reaction is

    (a) 0

    (b) 1

    (c) 2

    (d) 3

Ans.    (b)

Sol.    

    

42.    Consider the Hamiltonian , where and are parameters with appropriate dimensions, and q and p are the generalized coordinate and momentum, respectively. The corresponding Lagrangian is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

43.    For a given load resistance R_L = 4.7 ohm, the power transfer efficiencies of a dc voltage source and a dc current source with internal resistances R₁ and R₂, respectively, are equal. The product R₁R₂ in units of ohm² (rounded off to one decimal place) is ___________

Ans.    22.09

Sol.    For dc voltage source

    

    

    For dc current

    

    

    

44.    The ground state electronic configuration of the rare-earth ion (Nd³⁺) is [Pd] 4f³5s²5p⁶. Assuming LS coupling, the Landé g-factor of this ion is . The effective magnetic moment in units of Bohr magneton µ_B (rounded off to two decimal places) is __________

Ans.    3.62

Sol.    

    

45.    A projectile of mass 1 kg is launched at an angle of 30º from the horizontal direction at t = 0 and takes time T before hitting the ground. If its initial speed is 10 ms^–1, the value of the action integral for the entire flight in the units of kg m²s^–1 (rounded off to one decimal place) is ____________

    [Take g = 10 ms^–2]

Ans.    33.3

Sol.    

    

    

    L = 100t² – 100t + 50

    

46.    Let be a variable in the range . Now consider a funtion

        

    If its Fourier-series is written as , then the value of |C₃|² (rounded off to three decimal places) is ________________

Ans.    0.011

Sol.    

    

    

47.    Two spaceships A and B, each of the same rest length L, are moving in the same direction with speeds and , respectively, where c is the speed of light. As measured by B, the time taken by A to completely overtake B [see figure below] in units of L/c (to the nearest integer) is ______________

    

Ans.    5

Sol.    

    Kinematic equation is given by

    

48.    A radioactive element X has a half-life of 30 hours. It decays via alpha, beta and gamma emissions with the branching ratio for beta decay being 0.75. The partial half-life for beta decay in unit of hours is ______________

Ans.    40

Sol.    Branching ratio is the fraction of particle (here ) which decays by an individual decay mode with respect to the total number of particles which decays

    

49.    In a thermally insulated container, 0.01 kg of ice at 273 K is mixed with 0.1 kg of water at 300 K. Neglecting the specific heat of the container, the change in the entropy of the system in J/K on attaining thermal equilibrium (rounded off to two decimal places) is ____________

    (Specific heat of water is 4.2 kJ/kg-K and the latent heat of ice is 335 kJ/kg).

Ans.    1.03

Sol.    T_eq = 290.29 K    (Heat gain = Heat lost)

    

    

    

50.    Consider a system of three charges as shown in the figure below:

    For r = 10 m; = 60 degrees; q = 10^–6 Coulomb, and d = 10^–3 m, the electric dipole potential in volts (rounded off to three decimal places) at a point (r, ) is __________________

    [Use : ]

Ans.    0.045

Sol.    

    

51.    Consider two systems A and B each having two distinguishable particles. In both the systems, each particle can exist in states with energies 0, 1, 2 and 3 units with equal probability. The total energy of the combined system is 5 units. Assuming that the system A has energy 3 units and the system B has energy 2 units, the entropy of the combined system is k_B ln . The value of is ____________________

Ans.    12

Sol.    

    

52.    Electrons with spin in the z-direction are passed through a Stern-Gerlach (SG) set up with the magnetic field at = 60º from . The fraction of electrons that will emerge with their spin parallel to the magnetic field in the SG set up (rounded off to two decimal places) is _______________

    

Ans.    0.25

Sol.     state related to up state is

    The fraction of electrons that will emerge with their spin parallel to the magnetic field

    

53.    The Hamiltonian of a system is with << 1. the fourth order contribution to the ground state energy of H is . The value of (rounded off to three decimal places) is _______________

Ans.    0.125

Sol.     the eigen value of the Hamiltonian is

    The ground state is E_g =

    Taylor expansion of

    

54.    Two events, one on the earth and the other one on the Sun, occur simultaneously in the earth's frame. The time difference between the two events as seen by an observer in a spaceship moving with velocity 0.5c in the earth's frame along the line joining the earth to the Sun is where c is the speed of light. Given that light travels from the Sun to the earth in 8.3 minutes in the earth's frame, the value of in minutes (rounded off to two decimal places) is ________________

    (Take the earth's frame to be inertial and neglect the relative motion between the earth and the sun)

Ans.    4.77

Sol.    

    

55.    In a certain two-dimensional lattice, the energy dispersion of the electrons is

        

    where denotes the wave vector, a is the lattice constant and t is a constant in units of eV. In this lattice the effective mass tensor m_ij of electrons calculated at the center of the Brillouin zone has the form . The value of (rounded off to three decimal places) is ________________

Ans.    0.33

Sol.    Effective mass tensor matrix 4

    

    

    At the Brillouin zone centre i.e. at k_x = k_y = 0

    

    

    At the Brillouin zone centre i.e. at k_x = k_y = 0