1.    The eigenvalues of a Hermitian matrix are all

    (a) real

    (b) imaginary

    (c) of modulus one

    (d) real and positive

Ans.    (a)

Sol.    Eigenvalue of Hermitian matrix must be real.

2.    Which one of the following represents the 3p radial wave function of hydrogen atom? (a₀ is the Bohr radius)

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    3p radial wave function is

3.    Given the following table,

    

    Which one of the following correctly matches the experiments from Group I to their inferences in Group II?

    (a) P-2, Q-3, R-4, S-1

    (b) P-1, Q-3, R-2, S-4

    (c) P-3, Q-4, R-2, S-1

    (d) P-2, Q-1, R-4, S-3

Ans.    (c)

Sol.    The correct option is (c)

4.    In spherical polar coordinates , the unit vector at is

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    

    

    

5.    The scale factors corresponding to the covariant metric tensor g_ij in spherical polar coordinates are

    (a) 1, r², r² sin²

    (b) 1, r², sin²

    (c) 1, 1, 1

    (d) 1, r, r sin

Ans.    (d)

Sol.    The correct option is (d).

6.    In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?

    (a) Each normal coordinate has an eigen-frequency associated with it

    (b) The normal coordinates are orthogonal to one another

    (c) The normal coordinates are all independent

    (d) The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients

Ans.    (b)

Sol.    Normal coordinate must be independent. It is not necessary that it should orthogonal.

7.    For the given unit cells of a two dimensional square lattice, which option lists all the primitive cells?

    (a) (1) and (2)

    (b) (1), (2) and (3)

    (c) (1), (2), (3) and (4)

    (d) (1), (2), (3), (4) and (5)

Ans.    (c)

Sol.    For primitive cell, N_eff = 1

    In cell (1), (2), (3) and (4) N_eff = 1, these are primitive cell

    Whereas in cell (5), N_eff = 2, this is non-primitive cell.

8.    Among electric field , magnetic field , angular momentum , and vector potential , which is/are odd under parity (space inversion) operation?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

9.    The expression for the second overtone frequency in the vibrational absorption spectra of a diatomic molecule in terms of the harmonic frequency and anharmonicity constant x_e is

    (a) 2 (1 – x_e)

    (b) 2 (1 – 3x_e)

    (c) 3 (1 – 2x_e)

    (d) 3 (1 – 4x_e)

Ans.    (d)

Sol.    

    

10.    Match the physical effects and order of magnitude of their energy scales given below, where is fine structure constant; m_e and m_p are electron and proton mass, respectively.

    

    (a) P-3, Q-1, R-2, S-4

    (b) P-2, Q-3, R-1, S-4

    (c) P-4, Q-2, R-1, S-3

    (d) P-2, Q-4, R-1, S-3

Ans.    (c)

Sol.    

    

11.    The logic expression can be simplified to

    (a) A XOR C

    (b) A AND

    (c) 0

    (d) 1

Ans.    (a)

Sol.    

12.    At low temperatures (T), the specific heat of common metals is described by (with and as constants)

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

13.    In a 2-to-1 multiplexer as shown below, the output X = A₀ if C = 0, and X = A₁ if C = 1.

    Which one of the following is the correct implementation of this multiplexer?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Check option (a),

    

14.    The elementary particle is placed in the baryon decuplet, shown below, at

    (a) P

    (b) Q

    (c) R

    (d) S

Ans.    (c)

Sol.    

15.    The intrinsic/permanent electric dipole moment in the ground state of hydrogen atom is (a₀ is the Bohr radius)

    (a) –3ea₀

    (b) zero

    (c) ea₀

    (d) 3ea₀

Ans.    (b)

Sol.    For dipole moment energy is –eEr

16.    The high temperature magnetic susceptibility of solids having ions with magnetic moments can be described by with T as absolute temperature and as constant. The three behaviors i.e. paramagnetic, ferromagnetic and anti-ferromagnetic are described, respectively, by

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

17.    Which one of the following is an allowed electric dipole transition?

    (a) ¹S₀³S₁

    (b) ²P_3/2²D_5/2

    (c) ²D_5/2²P_1/2

    (d) ³P₀⁵D₀

Ans.    (b)

Sol.    For electric dipole transition

    

    only option (b) satisfies above selection rules.

18.    In the decay, , what is X?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    

    

19.    A spaceship is travelling with a velocity of 0.7c away from a space station. The spaceship ejects a probe with a velocity 0.59c opposite to its own velocity. A person in the space station would see the probe moving at a speed Xc, where the value of X is ________ (up to three decimal places).

Ans.    0.187c

Sol.    v = 0.7c, u'_x = –0.59c

    

    

20.    For an operational amplifier (ideal) circuit shown below,

    if V₁ = 1 V and V₂ = 2 V, the value of V₀ is _______ V (up to one decimal place).

Ans.    –3.6

Sol.    

    V₀ = – 2 – 1.6 = – 3.6V

21.    An infinitely long straight wire is carrying a steady current I. The ratio of magnetic energy density at distance r₁ to that at r₂(= 2r₁) from the wire is _________.

Ans.    4

Sol.    

22.    A light beam of intensity I₀ is falling normally on a surface. The surface absorbs 20% of the intensity and the rest is reflected. The radiation pressure on the surface is given by X I₀/c, where X is ________ (up to one decimal place). Here c is the speed of light.

Ans.    1.8

Sol.    

23.    The number of independent components of a general electromagnetic field tensor is _________.

Ans.    6

Sol.    In Cartesian co-ordinate, three independent coordinate for electric field (E_x, E_y, E_z) and three independent co-ordinate for magnetic field (B_x, B_y, B_z).

24.    If X is the dimensionality of a free electron gas, the energy (E) dependence of density of states is given by , where Y is _________.

Ans.    1

Sol.    

25.    For nucleus ¹⁶⁴Er, a = 2⁺ state is at 90 keV. Assuming ¹⁶⁴Er to be a rigid rotor, the energy of its 4⁺ state is _______ keV (up to one decimal place).

Ans.    300

Sol.    E_J = hcBJ (J + 1)    ________4⁺

    E₂₊ hc B 2(2 + 1) and E₄₊ = hc B 4(4 + 1)    ________2⁺

    

26.    Given and , which one of the following makes a complete set for a three dimensional real linear vector space?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Let A be the matrix formed by taking as column matrix i.e.,

    

    Since, |A| 0, hence form a three dimensional real vector space.

    Hence, option (d) is correct.

27.    An interstellar object has speed v at the point of its shortest distance R from a star of much larger mass M. Given v² = 2 GM/R, the trajectory of the object is

    (a) circle

    (b) ellipse

    (c) parabola

    (d) hyperbola

Ans.    (c)

Sol.    At shortest distance E =

    Since, mvR = J J² = m²v²R²

    Now, J² = m²2GMR = 2GMm²R    (Given that v² = )

    

    For Kepler's potential, if energy is zero, then the shape is parabola.

28.    A particle moves in one dimension under a potential V(x) = |x| with some non-zero total energy. Which one of the following best describes the particle trajectory in the phase space?

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

    

    

29.    Consider an infinitely long solenoid with N turns per unit length, radius R and carrying a current I(t) = cos , where is a constant and is the angular frequency. The magnitude of electric field at the surface of the solenoid is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

    

30.    A constant and uniform magnetic field pervades all space. Which one of the following is the correct choice for the vector potential in Coulomb gauge?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    Check option (c),

    

31.    If H is the Hamiltonian for a free particle with mass m, the commutator [x, [x, H]] is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    For free particle, potential is zero.

    

32.    A long straight wire, having radius a and resistance per unit length r, carries a current I. The magnitude and direction of the Poynting vector on the surface of the wire is

    (a) I²r/2a, perpendicular to axis of the wire and pointing inwards

    (b) I²r/2a, perpendicular to axis of the wire and pointing outwards

    (c) I²r/a, perpendicular to axis of the wire and pointing inwards

    (d) I²r/a, perpendicular to axis of the wire and pointing outwards

Ans.    (a)

Sol.    

33.    Three particles are to be distributed in four non-degenerate energy levels. The possible number of ways of distribution: (i) for distinguishable particles, and (ii) for identical Bosons, respectively, is

    (a) (i) 24, (ii) 4

    (b) (i) 24, (ii) 20

    (c) (i) 64, (ii) 20

    (d) (i) 64, (ii) 16

Ans.    (c)

Sol.    Number of particles, N = 3

    Number of state, g = 4

    For distinguishable particle, w = g^N = 4³ = 64

    

34.    The term symbol for the electronic ground state of oxygen atom is

    (a) ¹S₀

    (b) ¹D₂

    (c) ³P₀

    (d) ³P₂

Ans.    (d)

Sol.    O ; 1s², 2s², 2p⁴

    

    Here, S = 1, L = 2

    According to Hund's rule, for ground state energy

    J = (L + S) = 2

    ^{2S + 1}L_J = ³P₂

35.    The energy dispersion for electrons in one dimensional lattice with lattice parameter a is given by , where W and E₀ are constants. The effective mass of the electron near the bottom of the band is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

36.    Amongst electrical resistivity thermal conductivity specific heat (C), Young's modulus (Y), and magnetic susceptibility which quantities show a sharp change at the superconducting transition temperature?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    The correct option is (b).

37.    A quarter wave plate introduces a path difference of between the two components of polarization parallel and perpendicular to the optic axis. An electromagnetic wave with is incident normally on a quarter wave plate which has its optic axis making an angle 135º with the x-axis as shown.

    The emergent electromagnetic wave would be

    (a) elliptically polarized

    (b) circularly polarized

    (c) linearly polarized with polarization as that of incident wave

    (d) linearly polarized but with polarization at 90º to that of the incident wave

Ans.    (c)

Sol.    The correct option is (c).

38.    A p-doped semiconductor slab carries a current I = 100 mA in a magnetic field B = 0.2 T as shown. One measures V_y = 0.25 mV and V_x = 2 mV. The mobility of holes in the semiconductor is __________ m²V^–1s^–1 (up to two decimal places).

Ans.    1.55

Sol.    The answer is 1.55

39.    An n-channel FET having Gate-Source switch-off voltage V_GS(OFF) = –2 V is used to invert a 0 – 5 V square-wave signal as shown. The maximum allowed value of R would be __________ (up to two decimal places).

Ans.    0.70

Sol.    The answer is 0.70.

40.    Inside a large nucleus, a nucleon with mass 939 MeVc^–2 has Fermi momentum 1.40 fm^–1 at absolute zero temperature. Its velocity is Xc, where the value of X is ___________ (up to two decimal places).

    ( 197 MeV-fm)

Ans.    0.29

Sol.    Here, fermi-momentum or fermi-radius, k_F = 1.40 fm^–1 and hc = 197 MeV-fm

    Now, Fermi-velocity

        

41.    4 MeV -rays emitted by the de-excitation of ¹⁹F are attributed, assuming spherical symmetry, to the transition of protons from 1d_3/2 state to 1d_5/2 state. If the contribution of spin-orbit term to the total energy is written as , the magnitude of C is ___________ MeV (up to one decimal place).

Ans.    1.6

Sol.    

    

    

42.    An particle is emitted by a nucleus. Assuming the potential to be purely Coulombic beyond the point of separation, the height of the Coulomb barrier is _________ MeV (up to two decimal places).

    

Ans.    25.995

Sol.    The height of coulomb barrier for particle from

    

    

    And R = R₀A^1/3

    Here, we consider pure Coulombic interection

    

    

    V_C = 25.995 MeV

43.    For the transformation

        

    (where is a constant) to be canonical, the value of is ____________.

Ans.    2

Sol.    

    Since, [Q, P] = 1

    

    

44.    Given

        ,

    and boundary conditions f(0) = 1 and f(1) = 0, the value of f(0.5) is __________ (up to two decimal places).

Ans.    0.81

Sol.    

    Auxiliary equation is,

    (m² – 2m +1) = 0

    (m – 1)² = 0 m = 1, 1

    Hence, the solution is

    f(x) = (c₁ + c₂x)e^x

    using boundary condition,

    f(0) = c₁e⁰ c_z = 1        ...(i)

    f(1) = (c₁ + c₂)e = 0        ...(ii)

    From (i) and (ii), c₂ = –1

    Hence, f(x) = (1 – x)e^x f(0.5) = (1 – 0.5)e^0.5 = 0.81

45.    The absolute value of the integral

            ,

    over the circle |z – 1.5| = 1 in complex plane, is _________ (up to two decimal places).

Ans.    81.64

Sol.    

    Pole, z = 2, –2

    z = –2 is outside the center

    |–2 – 1.5| > 1 So, will not be considered

    

46.    A uniform circular disc of mass m and radius R is rotating with angular speed about an axis passing through its center and making an angle = 30º with the axis of the disc. if the kinetic energy of the disc is m²R², the value of is _________ (up to 2 decimal places).

Ans.    0.21

Sol.    The kinetic energy of the disc is,

    

    where is angular momentum and is angular velocity

    

    Hence, = 0.21

47.    The ground state energy of a particle of mass m in an infinite potential well is E₀. It changes to E₀(1 + × 10^–3), when there is a small potential bump of height and width a = L/100, as shown in the figure. The value of is ________ (up to two decimal places).

Ans.    0.81

Sol.    

    

    

    

48.    An electromagnetic plane wave is propagating with an intensity I = 1.0 × 10⁵ Wm^–2 in a medium with and µ = µ₀. The amplitude of the electric field inside the medium is _________ × 10³ Vm^–1 (up to one decimal place).

    

Ans.    6.6

Sol.    

    

49.    A microcanonical ensemble consists of 12 atoms with each taking either energy 0 state, or energy state. Both states are non-degenerate. If the total energy of this ensemble is 4, its entropy will be ________ k_B (up to one decimal place), where k_B is the Boltzmann constant.

Ans.    6.204

Sol.    The number of ways having total energy 4, out of 12 atoms is

    

    Hence, entropy, S = k_B lnw = k_B ln (495) = k_B (6.204) = 6.204 k_B

50.    A two-state quantum system has energy eigenvalues corresponding to the normalized states . At time t = 0, the system is in quantum state . The probability that the system will be in the same state at t = h/(6) is ____________ (up to two decimal places).

Ans.    0.25

Sol.    

    

    Now probability in same state

    

51.    An air-conditioner maintains the room temperature at 27ºC while the outside temperature is 47ºC. The heat conducted through the walls of the room from outside to inside due to temperature difference is 7000 W. The minimum work done by the compressor of the airconditioner per unit time is ________ W.

Ans.    466.67

Sol.    Q₂ + W = Q₁

    Coefficient of performance of refrigerator (AC) =

    Also, coefficient of performance of refrigerator, =

    

52.    Two solid spheres A and B have same emissivity. The radius of A is four times the radius of B, and temperature of A is twice the temperature of B. The ratio of the rate of heat radiated from A to that from B is _________.

Ans.    256

Sol.    

    R_A = 4R_B and T_A = 2T_B

    

53.    The partition function of an ensemble at a temperature T is , where k_B is the Boltzmann constant. The heat capacity of this ensemble at is X Nk_B, where the value of X is _________ (up to two decimal places).

Ans.    0.42

Sol.    The partition function, z =

    

    

54.    An atom in its singlet state is subjected to a magnetic field. The Zeeman splitting of its 650 nm spectral line is 0.03 nm. The magnitude of the field is ________ Tesla (up to two decimal places).

    (e = 1.60 × 10^–19 C, m_e = 9.11 × 10^–31 kg, c = 3.0 × 10⁸ ms^–1)

Ans.    1.52

Sol.    

    

55.    The quantum effects in an ideal gas become important below a certain temperature T_Q when de Broglie wavelength corresponding to the root mean square thermal speed becomes equal to the inter-atomic separation. For such a gas of atoms of mass 2 × 10^–26 kg and number density 6.4 × 10²⁵ m^–3, T_Q = __________ × 10^–3 K (up to one decimal place).

Ans.    84.2

Sol.    

    

    

    = 0.0842 K = 84.2 × 10^–3 K