1.    A satellite is moving in a circular orbit around the Earth. If T, V and E are its average kinetic, average potential and total energies, respectively, then which one of the following options is correct ?

    (a) V = –2T; E = –T

    (b) V = –T; E = 0

    (c)

    (d)

Ans.    (a)

Sol.    According to Virial theorem,

    

    The potential energy of the satellite V(r) < r^–1

    

    V = – 2T

    

2.    The lattice parameters a, b, c of an orthorhombic crystal are related by a = 2b = 3c. In units of a, the interplanar separation between the (110) planes is_______ (upto three decimal places).

Ans.    0.477 a

Sol.    Lattice parameters a, b, c have relation a = 2b = 3c  

    and the miller indices of the plane (hkl) = (110) i.e., h 1. k = l, l = 0.

    Therefore, interplanar seperation.

    

3.    Consider w = f(z) = u(x, y) + iv(x, y) to be an analytic function in a domain D. Which one of the following options is not correct ?

    (a) u(x, y) satisfies Laplace equation in D

    (b) v(x, y) satisfies Laplace equation in D

    (c) is dependent on the choice of the contour between z₁ and z₂ in D

    (d) f(z) can be Taylor expanded in D

Ans.    (c)

Sol.    f(z) = u(x, y) + iv (x,y)

    The real part part u(x, y) and imaginary part v(x, y) of a complex analytic function f(z) are harmonic functions i.e., they will satisfy Laplace's equation.

    Since, f(z) is analytic in the domain D, then (f)z can be Taylor expanded and the value of be independent of the choice of the contour.

4.    Let be the angular and linear momentum operators, respectively, for a particle. The commutator [L_x, p_y] gives

    (a)

    (b) 0

    (c)

    (d)

Ans.    (d)

Sol.    

5.    The dispersion relation for photons in a one dimensional monatomic Bravais lattice with lattice spacing a and consisting of ions of masses M is given by, , where is the frequency of oscillation, k is the wavevector and C is the spring constant. For the long wavelength modes , the ratio of the phase velocity to the group velocity is_______

Ans.    1

Sol.    The –k relation for monatomic lattice is given by

    

    Here, we have taken modulus of sin because cannot be negative.

    For long wavelength modes

    

    Ratio of phase velocity to the group velocity

    Correct answer is (1).

6.    For a black body radiation in a cavity, photons are created and annihilated freely as a result of emission and absorption by the walls of the cavity. This is because

    (a) the chemical potential of the photons is zero

    (b) photons obey Pauli exclusion principle

    (c) photons are spin-1 particles

    (d) the entropy of the photons is very large

Ans.    (a)

Sol.    Since photon has zero chemical potential the number of the photons cannot be conserved. Thus, photons can be created and annihilated.

7.    Four forces are given below in Cartesian and spherical polar coordinates.

    

    where K is a constant. Identify the correct option.

    (a) (iii) and (iv) are conservative but (i) and (ii) are not

    (b) (i) and (ii) are conservative but (iii) and (iv) are not

    (c) (ii) and (iii) are conservative but (i) and (iv) are not

    (d) (i) and (iii) are conservative but (ii) and (iv) are not

Ans.    (d)

Sol.    

    

    

8.    The value of is_______ (upto one decimal place)

Ans.    1.67

Sol.    

9.    The mean kinetic energy of a nucleon in a nucleus of atomic weight A varies as Aⁿ, where n is_______ (upto two decimal places)

Ans.    –0.67

Sol.    Total kinetic energy (K.E.) ofTY–neutrons and Z-protons is given by

    

    

10.    In Bose-Einstein condensates, the particles

    (a) have strong interparticle attraction

    (b) condense in real space

    (c) have overlapping wavefunctions

    (d) have large and positive chemical potential

Ans.    (c)

Sol.    Since the interparticle distance between the bosons is comparable to their de–Broglie wavelengths, their wavefuntions overlap.

11.    A beam of X - ray of intensity I₀ is incident normally on a metal sheet of thickness 2 mm. The intensity of the transmitted beam is 0.025 I₀. The linear absorption coefficient of the metal sheet (in m^–1) is __________ (upto one decimal place).

Ans.    1844.4

Sol.    If α is the linear absorption coefficient of the metal sheet then the transmitted intensity is given by

    

    

12.    In a Hall effect experiment, the hall voltage for an intrinsic semiconductor is negative. This is because (symbols carry usual meaning)

    (a)

    (b) n > p

    (c) µ_* > µ_h

    (d)

Ans.    (d)

Sol.    The hall coefficient of intrinsic semiconductor is given by

    

    Given hall voltage is negative,

    

13.    The Pauli matrices for three spin- particles are , respectively. The dimension of the Hilbert space required to define an operator is_______

Ans.    8

Sol.    The required number of dimensions

    = (25 + 1)ⁿ, where 5 is the spin of the particle and n is the number of particle.

    

14.    The decay is forbidden, because it violates

    (a) momentum and lepton number conservations

    (b) baryon and lepton number conservations

    (c) angular momentum conservation

    (d) lepton number conservation

Ans.    (d)

Sol.    

    

    Electronic and muonic leptqn number should be conserved separately. Here both are not conserved. Correct option is (d)

15.    The space between two plates of a capacitor carrying charges +Q and –Q is filled with two different dielectric materials, as shown in the figure. Across the interface of the two dielectric materials, which one of the following statements is correct ?

    (a) and are continuous

    (b) is continuous and is discontinuous

    (c) is continuous and is discontinuous

    (d) and are discontinuous

Ans.    (c)

Sol.    Suppose the line AB is the interface between two dielectrics.

    We can use the boundary condition for and D

    Across interface, parallel component of is continuous while perpendicular component suffers a discontinuity of where is the permitivity of medium.

    

    

16.    Given that magnetic flux through the closed loop PQRSP is . If along PQR, the value of along PSR is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

17.    A point charge is placed between two semi-infinite conducting plates which are inclined at an angle of 30° with respect to each other. The number of image charges is_______

Ans.    11

Sol.    The number of image charges

    

18.    Consider a complex function . Which one of the following statements is correct ?

    (a) f(z) has simple poles at

    (b) f(z) has a second order pole at

    (c) f(z) has infinite number of second order poles

    (d) f(z) has all simple poles

Ans.    (b)

Sol.    

    

    Highest negative power term in the above Laurrent series expansion is ]

    Therefore, f(z) has a pole of order 2 at z = 4 –

19.    The energy dependence of the density of states for a two dimensional non-relativistic electron gas is given by, g(E) = CEⁿ, where C is constant. The value of n is __________

Ans.    0

Sol.    The density of states in 2-D is given by

    

    Since electron spin degeneracy is 2 so we should multiply density of states by a factor of 2.

    Therefore,

    On comparing with g (E) = CEⁿ, we have n = 0

20.    In an inertial farme S, two events A and B take place at and , respectively. The times at which these events take place in a frame S′ moving with a velocity with respect to S are given by

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    According to Lorentz transform

    

21.    In the given circuit, the voltage across the source resistor is 1 V. The drain voltage (in V) is _______

Ans.    15

Sol.    Given; Source voltage

    V_s = 1V

    To find Drain voltage V_D

    Applying KVL at input

    V_GS + 1 = 0

    V_GS = – Volt

    KVL at output

    25 = 5 × 1_D + V_D

    V_D = 25 – 5 × 1_D  put  1_D = 2 mA

    V_D = 15 Volt

22.    If , then

    (a) f and g are differentiable everywhere

    (b) f is differentiable everywhere but g is not

    (c) g is differentiable everywhere but f is not

    (d) g is discontinuous at x = 0

Ans.    (b)

Sol.    This question can be solved easily by plotting f(x) and g(x)

    

    

    

    

    From graph, it is clear. g(x) is nofdifferentiable at x =

    Correct option is (b)

23.    Consider a system of N non-interacting spin- particles, each having a magnetic moment µ, is in a magnetic field . If E is the total energy of the system, the number of accessible microstates is given by

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    Let N₊ = Number of particles with spin (or up)

    and N_– = Number of particles with spin (or down).

    So, we have

    N = N₊ + N_–                                        ... (i)

    Also, the total energy is

    

    Solving (i) and (ii), we have

    

    Total number of microstates (fi) when n particles are distributed in two states n₊ and n_– is

    

24.    Which one of the following does not represent an exclusive or operation for inputs A and B ?

    (a)

    (b)

    (c)

    (d) (A + B)AB

Ans.    (d)

Sol.    

    

    

    (d)    (A + B) AB

        = AAB + ABB = AB + AB = AB

        AND logic - gate

    Note: A + A = A

25.    An operator for a spin- particle is given by denotes Pauli matrices and is a constant. The eigenvalue of are

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    Eigenvalue equation:

    

26.    Match the phrases in Group I and Group II and identify the correct option.

         Group I                         Group II

    (P)     Electron spin resonance (ESR)            (i)     radio frequency

    (Q)     Nuclear magnetic resonance (NMR)            (ii) visible range frequency

    (R)     Transition between vibrational states of a molecule    (iii) microwave frequency

    (S)     Electronic transition                (iv) far-infrared range

    (a) (P-i), (Q-ii), (R-iii), (S-iv)

    (b) (P-ii), (Q-i), (R-iv), (S-iii)

    (c) (P-iii), (Q-iv), (R-i), (S-ii)

    (d) (P-iii), (Q-i), (R-iv), (S-ii)

Ans.    (d)

Sol.    P-(iii): ESR (electron spin resonance) lies in GHz [8–10 GHz] frequency range. Thus, it lies in microwave range.

    Q-(i): NMR (nuclear magnetic resonance) lies in [100 – 1000] MHz frequency range. Thus, it is in radio frequency range.

    R-(iv): Transition between vibrational states of a molecule lies in frequency range (10¹² to 10¹⁴ Hz). Thus, it lies in farinfrared range.

    S-(ii): Electronic transition lies in frequency range 10¹⁴ to 10¹⁵ Hz. Thus, it lies in visible range.

27.    The entropy of a gas containing N particles enclosed in a volume V is given by , where E is the total energy, a is a constant and k_B is the Boltzmann constant. The chemical potential µ of the system at a temperature T is given by

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    From first law of thermodynamics,

    TdS = dE + PdV– µdN

    At constant E and V,

    

28.    The atomic masses of and neutron are 151.921749, 151.919756, 1.007825 and 1.008665 in atomic mass units (amu), respectively. Using the above information, the Q-value of the reaction amu (upto three decimal places)

Ans.    2.833

Sol.    Q = (M_En + M_n) – (M_sm + M_p)

    = (151.921749 + 1.008665) – (151.919756 + 1.007825) = 2.833 × 10^–3 a.m.u.

29.    A particle with rest mass M is at rest and decays into two particles of equal rest masses which move along the z-axis. Their velocities are given by

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    According to energy conservation

    

    Since, initial velocity was zero and two parts have same mass.

    So, (for momentum to be conserved)

    

30.    The band gap of an intrinsic semiconductor is E_g = 0.72 eV and . At 300K, the Fermi level with respect to the edge of the valence band (in eV) is at ________________ (upto three decimal places) k_B = 1.38 × 10^–23 JK^–1.

Ans.    0.395

Sol.    

    

31.    A charge –q is distributed uniformly over a sphere, with a positive charge q at its center in (i). Also in (ii), a charge –q is distributed uniformly over an ellipsoid with a positive charge q at its center. With respect to the origin of the coordinate system, which one of the following statements is correct ?

    

    (a) The dipole moment is zero in both (i) and (ii)

    (b) The dipole moment is non-zero in (i) but zero in (ii)

    (c) The dipole moment is zero in (i) but non-zero in (ii)

    (d) The dipole moment is non-zero in both (i) and (ii)

Ans.    (a)

Sol.    Total charge in the given figures (i) and (ii) are zero.

    Therefore, value of dipole moment is independent of origin.

    The charge distribution in (i) and (ii) are both symmetric about origin.

    Dipole moment due to symmetric charge distribution is zero.

    Therefore, dipole moment is zero in both (i) and (ii).

32.    The number of permitted transitions from ²P_3/2²S_1/2 in the presence of a weak magnetic field is______

Ans.    6

Sol.    For ²P_3/2²S_1/2:

    In presence of an external weak field, the state ²S_1/2 will split into two states and ²p_3/2 state will split into four state .

    Selection rule;

    Thus, there will be total six transitions permitted in

    Correct answer is (6).

33.    A particle is confined in a box of length L as shown below.

    If the potential V₀ is treated as a perturbation, including the first order correction, the ground state energy is

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    

    First order energy correction

    

34.    In the given circuit, if the open loop gain A = 10⁵, the feedback configuration and the closed loop gain A_f are

    (a) series-shunt, A_f = 9

    (b) series-series, A_f = 10

    (c) series-shunt, A_f = 10

    (d) shunt-shunt, A_f = 10

Ans.    (c)

Sol.    

    

    Non inverting amplifier is an example of voltage series topology.

    Voltage series OR series-shunt

    Correct option is (c)

35.    A plane wave after passing through an optical element emerges as , where k and are the wavevector and the angular frequency, respectively. The optical element is

    (a) quarter wave plate

    (b) half wave plate

    (c) polarizer

    (d) Faraday rotator

Ans.    (b)

Sol.    

    So, phase introduce by the optical system =

36.    A particle of mass 0.01 kg falls freely in the earth's gravitational field with an initial velocity v(0) = 10 ms^–1. If the air exerts a functional force of the form, f = –kv, then for k = 0.05 Nm^–1s, the velocity (in ms^–1) at time t = 0.2 s is_______ (upto two decimal places)

    (use g = 10 ms^–2 and e = 2.72)

Ans.    4.9

Sol.    

        At t = 0, v = 10 ms^–1

    

    

37.    The Lagrangian for a particle of mass m at a position moving with a velocity is given by , where V(r) is a potential and C is a constant. If is the canonical momentum, then its Hamiltonian is given by

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

38.    A long solenoid is embedded in a conducting medium and is insulated from the medium. If the current through the solenoid is increased at a constant rate, the induced current in the medium as a function of the radial distance r from the axis of the solenoid is proportional to

    (a) r² inside the solenoid and outside

    (b) r inside the solenoid and outside

    (c) r² inside the solenoid and outside

    (d) r inside the solenoid and outside

Ans.    (d)

Sol.    According to Maxwell's equation,

    

    For inside point of the solenoid

    

    For outside point of the solenoid

    

    

    

39.    In the nuclear shell model, the potential is modeled as . The correct spin-parity and isospin assignments for the ground state of ¹³C is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    We have nucleus, ¹³C₆

    Z = 6 (even)

    

    Spin parity of ground state is

    Isospin assignment i.e. third component of isospin

40.    Which one of the following represents the electron occupancy for a superconductor in its normal and superconducting states?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    The superconductor behaves like a normal conductor above transition temperature (T_C) and below T_c, it exhibits superconductivity.

    In normal state the electrons will have integral spin and

    will follow Fermi-Dirac distribution function.

    

    

    Therefore, f(E) versus E graph (T > 0K) will be

    In superconducting state, electrons form cooper pairs with spin = 0.

    Hence these electrons in superconducting state will follow Bose-Einstein distribution function.

    

    

    Therefore, f(E) versus E graph (T > 0K) will be

    Hence, correct option is (b).

41.    In a rigid-rotator of mass M, if the energy of the first excited state is 1 meV, then the fourth excited state energy (in meV) is_______

Ans.    10

Sol.    Energy of a rigid rotator is

    

    where J = 0, 1, 2,...

    Energy of first excited state

    

    Energy of fourth excited state

    

42.    The binding energy per molecule of NaCl (lattice parameter is 0.563 nm) is 7.95 eV. The repulsive term of the potential is of the form , where K is a constant. The value of the Madelung constant is ____________(upto three decimal places)

    (Electron charge e = 1.6 × 10^–19 C; ε₀ = 8.854 × 10^–12 C² N^–1m^–2)

Ans.    –1.750

Sol.    The cohesive energy of NaCl can be given by:

    

    

    

    ( is positive quantity)

43.    The Hamiltonian for a system of two particles of masses m₁ and m₂ at having velocities is given by , where C is a constant. Which one of the following statements is correct ?

    (a) The total energy and total momentum are conserved

    (b) Only the total energy is conserved

    (c) The total energy and the z-component of the total angular momentum are conserved

    (d) The total energy and total angular momentum are conserved

Ans.    (b)

Sol.    Since, the given Hamiltonian does not depends on time explicitly. So, total energy of the system will be conserved.

44.    Given that the Fermi energy of gold is 5.54 eV, the number density of electrons is _______________×10²⁸ m^–3 (upto one decimal place).

    (Mass of electron = 9.11×10^–21 kg; h = 6.626×10^–24J.s; 1 eV = 1.6×10^–19 J)

Ans.    5.9

Sol.    

    

    

    

45.    Suppose a linear harmonic oscillator of frequency ω and mass m is in the state at t = 0 where are the ground and the first excited states, respectively. The value of in the units of t = 0 is_______

Ans.    0

Sol.    

    We know that,

    

    

46.    Consider the motion of the Sun with respect to the rotation of the Earth about its axis. If denote the centrifugal and the Coriolis forces, respectively, acting on the Sun, then

    (a) is radially outward and

    (b) is radially inward and

    (c) is radially outward and

    (d) is radially outward and

Ans.    (d)

Sol.    Centrifugal force, F_c = – is radially outword

    Coriolis force, F_co = – = – = 2F.

47.    A function y(z) satisfies the ordinary differential equation , where m = 0, 1, 2, 3, .... . Consider the four statements P, Q, R, S as given below.

    P : z^m and z^–m are linearly independent solutions for all values of m

    Q : z^m and z^–m are linearly independent solutions for all values of m > 0

    R : z and 1 are linearly independent solutions for m = 0

    S : z^m and ln z are linearly independent solutions for all values of m

    The correct option for the combination of valid statement is

    (a) P, R and S only

    (b) P and R only

    (c) Q and R only

    (d) R and S only

Ans.    (c)

Sol.    

    

    Trial solution y = C.^kx, putting in equation (iii)

    

    So, y = C₁e^mx + C₂e^–mx = C₁z^m + C₁z^–m

    Therefore, z^m and z^–m are linearly independent solutions for m > 0.

    

    Therefore, 1 and lnz are linearly independent solutions for m = 0.

48.    The average energy U of a one dimensional quantum oscillator of frequency ω and in contact with a heat bath at temperature T is given by

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    The energy of a one-dimensional quantum oscillator with frequency is

    

    The partition function is

    

    

    

    The average energy U is

    

49.    Consider a system of eight non-interacting, identical quantum particles of spin- in a one dimensional box of length L. The minimum excitation energy of the system, in units of is_______

Ans.    5

Sol.    The number of degeneracy including spin is

    Therefore, the minimum exitation energy of the system

    

    

50.    In the simple current source shown in the figure, Q₁ and Q₂ are identical transistors with current gain = 100 amd V_BE = 0.7V

    The current I₀ (in mA) is_______(upto two decimal places)

Ans.    5.74

Sol.    • Given figure is a current mirror circuit. In this circuit output current is forced to be equal to input current.

    • Output current is mirror image of input current.

    • It is basically voltage to current converter

    

    Given:

    •     = 100

    •    V_BE = 100

    •    Here, Q₁ and Q₂ are identical transistor.

    

    

    

    

    Put the value of I_C1, I_B1, I_B2 in equation (1)

    

    Put the value of I_ref and

    

51.    The Heaviside function is defined as and its Fourier transform is given by –2i/. The Fourier transform of is

    (a)

    (b)

    (c)

    (d) 0

Ans.    (a)

Sol.    Shifting property:

    

    

52.    Consider the circuit shown in the figure, where RC = 1. For an input signal V_i shown below, choose the correct V₀ from the options:

    

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    It is a differentiator circuit

    As ideal op-amp has infinite input resistance. So, there will be no current passing through op-amp. So, by applying KCL at node A.

    

    So, output signal can be obtained by differentiating the input signal.

    Note: The circuit given is an inverting differentiator. (Just differentiate the given input signal)

53.    Let the Hamiltonian for two spin- particles of equal masses m, momenta and positions be , where denote the corresponding Pauli matrices, . If the ground state has net spin zero, then the energy (in eV) is_______

Ans.    –0.2

Sol.    

    

    Energy eigenvalues are

    

    For ground state, (n₁ = 0, n₂ = 0, s = 0)

    

54.    The excitation wavelength of laser in a Raman effect experiment is 546 nm. If the Stokes line is observed at 552 nm, then the wave number of the anti-Stokes line (in cm^–1) is_______

Ans.    18514

Sol.    The wave number of anti-stokes line is given by

    

    

    

    = 0.003663 × 10⁷ – 0.0018116 × 10⁷

    = 36630 – 18116percm =18514 per cm

55.    A monochromatic plane wave (wavelength = 600 nm) is incident normally on a diffraction grating giving rise to a plane wave in the first order of diffraction. Here . The period (in µm) of the diffraction grating is_______(upto one decimal place)

Ans.    1.2

Sol.    The condition for n^th order maxima is d sin

    For first order diffraction n = 1