1.    The unit vector perpendicular to the surface x² + y² + z² = 3 at the point (1, 1, 1) is

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Unit vector perpendicular to the surface.

    

2.    Which one of the following quantities is invariant under Lorentz transformation?

    (a) Charge density

    (b) Charge

    (c) Current

    (d) Electric field

Ans.    (b)

Sol.    Change in conserved for isolated system so it is relativistically invariant.

3.    The number of normal Zeeman splitting components of transition is

    (a) 3

    (b) 4

    (c) 8

    (d) 9

Ans.    (a)

Sol.    ¹D and ¹P is singlet state, because multiplicity 2s + 1 = 1 s = 0

    Therefore, total angular momentum J = L + S is equal to the total angular momentum L.

    Each energy level splits 2L + 1 levels as shown in figure below.

Selection rule: = 0, + 1

    Energy of each level is equal . Because of the uniform splitting of the levels. There are only three different transition energies

     corresponding to = + 1, 0, –1 respectively.

    The energy separation between two neighboring levels in P and D state are equal.

    Thus, these will be only 3 normal Zeeman splitting components of ¹P ¹D transition.    

4.    If the half-life of an elementary particle moving with speed 0.9c in the laboratory frame is 5×10^–8s, then the proper half-life is _____×10^–8 s. (c = 3×10⁸ m/s)

Ans.    2.18

Sol.    

    

5.    An unpolarized light wave is incident from air on a glass surface at the Brewster angle. The angle between the reflected and the refracted wave is

    (a) 0º

    (b) 45º

    (c) 90º

    (d) 120º

Ans.    (c)

Sol.    When light is incident at Brewster angle, then the reflected light and the refracted light make 90° angle with each other.

6.    Two masses m and 3m are attached to the two ends of a massless spring with force constant K. If m = 100g and K = 0.3 N/m, then the natural angular frequency of oscillation is _______ Hz.

Ans.    0.32

Sol.    

    

7.    The electric field of a uniform plane wave propagating in a dielectric, non-conducting medium is given by,

        

    The phase velocity of the wave is _________×10⁸ m/s.

Ans.    1.5

Sol.    

    

8.    The matrix is

    (a) orthogonal

    (b) symmetric

    (c) anti-symmetric

    (d) unitary

Ans.    (d)

Sol.    

    So, A is an unitary matrix

9.    The recoil momemtum of an atom is p_A when it emits an infrared photon of wavelength 1500 nm, and it is p_B when it emits a photon of visible wavelength 500 nm. The ratio is .

    (a) 1 : 1

    (b)

    (c) 1 : 3

    (d) 3 : 2

Ans.    (c)

Sol.    

    

10.    For a gas under isothermal conditions, its pressure P varies with volume V as . The bulk modulus B is proportional to

    (a) V^–1/2

    (b) V^–2/3

    (c) V^–3/5

    (d) V^–5/3

Ans.    (d)

Sol.    

    Where C is a constant of proportionality.    

    Bulk modulus is defined as

    

11.    Which one of the following high energy processes is allowed by conservation laws?

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    Therefore, it is not allowed reaction Baryon number is not conserved.

    

    Therefore, it is an allowed reaction.

    

    Therefore, is it not allowed reaction because electronic lepton number is not conserved

    

    It is not allowed reaction because, charge electronic and moronic lepton numbers are not conserved.

12.    The length element ds of an arc is given by, . The metric tensor g_if is

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    

    

13.    The ground state and the first excited state wave functions of a one dimensional infinite potential well are and , respectively. When two spin-up electrons are placed in this potential, which one of the following, with x₁ and x₂ denoting the position of the two electrons, correctly represents the space part of the ground state wave function of the system?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    Two spin up fermion cannot occupy the same quantum state the spin wave function is symmetric. So, the space part wave function must be anti-symmetric, since the total wave function of a fermion is anti-symmetric.

    

14.    If the vector potential

        

    satisfies the Coulomb gauge, the value of the constant is _________________

Ans.    1

Sol.    The Coulomb gauge condition is

    

    

15.    At a given temperature, T, the average energy per particle of a non-interacting gas of two-dimensional classical harmoic oscillator is _____________k_BT.

Ans.    2

Sol.    The Hamiltonian of a classical oscillator is given by

    

    Since we have four quadratic terms in the Hamiltonian and according to the law of equipartition of energy each would contribute to the average energy.

    So average energy per particle

    

16.    Which one of the following is a fermion?

    (a) particle

    (b) ₄Be⁷ nucleus

    (c) hydrogen atom

    (d) deuteron

Ans.    (b)

Sol.    Fermions have integral spin. Proton electron, neutron are fermions and have 1/2-spin.

    

    

17.    Which one of the following three-quark states (qqqq), denoted by X, cannot be a possible baryon? The corresponding electric charge is indicated in the superscript

    (a) X⁺⁺

    (b) X⁺

    (c) X^–

    (d) X^– –

Ans.    (d)

Sol.    Baryons have charge +2, +1, –1 but donot have –2 charge state.

    Therefore, X^— does not represent a Baryon.    

18.    The Hamilton's canonical equations of motion in terms of Poisson Brackets are

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    According to Poisson's equation

    

    

    

19.    The Miller indices of a plane passing through the three points having coordinates (0, 0, 1), (1, 0, 0), are

    (a) (2 1 2)

    (b) (1 1 1)

    (c) (1 2 1)

    (d) (2 1 1)

Ans.    (a)

Sol.    Let the plane is , where OA, OB and OC are intercepts on x, y and z-axis respectively.

 

    

    

20.    The plot of specific heat versus temperature across the superconducting transition temperature (T_C) is most appropriately represented by

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    The specific heat ^. C = C_el. + C_lattice

    Since, (C_el.)_{superconductor}, where a is a constant and is energy gap.

    i.e., the specific heat of a superconductor is discontinuous at transition temperature.

    Therefore, the electronic specific of a metal in the superconducting state varies with temperature in an exponential manner.    

21.    If is the orbital angular momentum and is the spin angular momentum, then does not commute with

    (a) S_z

    (b) L²

    (c) S²

    (d)

Ans.    (d)

Sol.    Here is spatial operators and in spin operator,

    So, L² will commute with and S² will commute with     

    And also = L² + 2 will commute with

    Since, S_z does not commute with S_x and S_y. So, S_z will not commute with .    

22.    The energy, for band electrons as a function of the wave vector, k in the first Brillouin zone of a one dimensional monatomic lattice is shown as (a is latticec constant)

    (a)

    (b)

    (c)

    (d)

Ans.    (b)

Sol.    The velocity v can given by

    

    This shows that, for a free electron, v is proportional to k. However, in the band theory, E_k is generally not proportional to k. The variation of E with k based on the band theory is shown in figure. (below).    

    

    This graph shows the slope dE/dk of the E(k) curve is not constant but changes with k. Using the curve and employing equation (ii), one can obtain v versus k as shown in figure (below).

    This curve indicates that the velocity of the electron is zero for k = 2 and , where the slope dE/dk is zero. i.e., at the top and bottom of the energy band (first Brillion zone). For k = k₀, where k₀ corresponds to the inflection point of E(k) curve, the absolute the value of the velocity attains a maximum value equal to free electron velocity. Beyond this point, the velocity decreases with increases in energy.

23.    For a free electron gas in two dimensions, the variation of the density of states, N(E) as a function of energy E, is the best represented by

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    Since the density of states, N(E) is independent of energy (E) for a free electron gas in two dimensions.

    Therefore, N(E) versus E graph can be represented as

    

    

24.    The input given to be an ideal OP-AMP integrator circuit is

    The correct output of the integrator circuit is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

25.    The minimum number of flip-flops required to construct a mod-75 counter is _________

Ans.    7

Sol.    Given Modulus = 75

    M < 2ⁿ            n is number of flip-plops

    75 < 2ⁿ        for    n = 6  Hence (n = 6 not possible)

    n = 7    

26.    A bead of mass 'm' can slide without friction along a massless rod kept at 45º with the vertical as shown in the figure. The rod is rotating about the vertical axis with a constant angular speed . At any instant, r is the distance of the bead from the origin. The momentum conjugate to 'r' is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    

27.    An electron in the ground state of the hydrogen atom has the wave function

        

    where a₀ is constant. The expectation value of the operator , where is:

    (Hint: )

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    

    

    

28.    For Nickel, the number density is 8×10²³ atoms/cm³ and electronic configuration is 1s²2s²2p⁶3s²3p⁶3d⁸4s². The value of the saturation magnetization of Nickel in its ferromagnetic states is ________ × 10⁹ A/m.

    (Given the value of Bohr magneton µ_B = 9.21 × 10^–21 Am²)

Ans.    4.4

Sol.    For Nickel, number density, n = 8 × 10²³ atom/cm³

    And its electronic configuration = 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁸ 4s²

    µ_B= 9.21 × 10^–21 Am^–2    

    n = 8 × 10²⁹/m³

    Magnetic moment of Ni = 0.6_B    (can be calculated using concept of atomc physics)

    Saturn magenetization = Nµ = 8 × 10²⁹ × 0.6 × 9.2 × 10^–21

    Correct option is (4.4)    

29.    A particle of mass 'm' is in a potential given by

        

    where 'a' and 'r₀' are positive constants. When disturbed slightly from its stable equilibrium position, it undergoes a simple harmonic oscillation. The time period of oscillation is

    (a)

    (b)

    (c)

    (d)

Ans.    (a)

Sol.    

    Time period of oscillation is given as

    

    At equilibrium position (r = r`)

    Putting r` = r₀

30.    The donor concentration in a sample of n-type silicon is increased by a factor of 100. The shift in the position of the Fermi level at 300K, assuming the sample to be non degenerate is ________ meV.

    (k_BT = 25 meV at 300K)

Ans.    115

Sol.    At T = 300 K, k_BT = 25 meV

    Let initial donor concentration and final donor concentration     

    

    

    = 2 × 2.303 × 25 meV = 115 meV

31.    A particle of mass 'm' is subjected to a potential

    

    The state with energy is g-fold degenerate. The value of 'g' is ____________________

Ans.    4

Sol.    The energy of the two dimensional harmonic oscillator.

    

    

    

32.    A hydrogen atom is in the state

    

    where denote the principal, orbital and magnetic quantum numbers, respectively. If is the angular momentum operator, the average value of is ______________ .

Ans.    2

Sol.    

    

33.    A planet of mass 'm' moves in a circular orbit of radius r₀ in the gravitational potential where 'k' is a positive constant. The orbital angular momentum of the planet is

    (a) 2r₀km

    (b)

    (c) r₀km

    (d)

Ans.    (d)

Sol.    

    

34.    The moment of inertia of a rigid diatomic molecule A is 6 times that of another rigid diatomic molecule B. If the rotational energies of the two molecules are equal, then the corresponding values of the rotational quantum numbers J_A and J_B are

    (a) J_A = 2, J_B = 1

    (b) J_A = 3, J_B = 1

    (c) J_A = 5, J_B = 0

    (d) J_A = 6, J_B = 1

Ans.    (b)

Sol.    Rotational energy of the rigid diatomic molecules is given by

    

    It terms of wave number unit,

    

    where, J is a rotational quantum number.

    I_A and I_B is moment of inertia of rigid rotator A and B respectively.

    

    

    This equation is satisfied for    

    J_A = 3 and J_B = 1    

35.    The value of the integral

    

    where C is the circle |z| = 4, is

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    Condition of singularity e^z + 1 = 0    

    

    

    

36.    A ray of light inside Region 1 in the xy-plane is incident at the semicircle boundary that carries no free charges. The electric field at the point in plane polar coordinates is , where and are the unit vectors. The emerging ray in Region 2 has the electric field parallel to x-axis. If and are the dielectric constants of Region 1 and Region 2 respectively, then is ______________

Ans.    2.33

Sol.    Given : Electric field in region 1 is

    

    According to boundary condition,

    

    

    

37.    The solution of the differential equation

    

    subject to the boundary conditions is

    (a) cos t + sin t

    (b) cosh t + sinh t

    (c) cos t – sin t

    (d) cosh t – sinh t

Ans.    (d)

Sol.    

    Assume, the trial solution to be y = c.e

    

    So, y = Ae^t + Be^–t    

    Putting y(t = 0) = 1, we get A + B = 1

    Putting y(t = ) = 8, we get A = 0, B = 1

    Therefore, y(t) = e^–t = cosh t – sin h t    

38.    Given that the linear transformation of a generalized coordinate 'q' and the corresponding momentum p,

    Q = q + 4ap

    p = q + 2p

    is canonical, the value of the constant 'a' is ___________

Ans.    0.25

Sol.    For canonical transformation

    Poisson Bracket {Q, P}_{q, p} = 1

    

    

39.    The value of the magnetic field required to maintain non-relativistic protons of energy 1 MeV in a circular orbit of radius 100 mm is ________ Tesla.

    (Given: m_p = 1.67 × 10^–27 kg, e = 1.6 × 10^–19 C)

Ans.    1.4450

Sol.    Kinetic energy = 1 MeV

    

    

40.    For a system of two bosons, each of which can occupy any of the two energy levels 0 and , the mean energy of the system at a temperature T with is given by

    (a)

    (b)

    (c)

    (d)

Ans.

Sol.    Two bosons particles can be distributed as

    

    So the partition function is

    

    The mean energy is

    

    None of the option are correct.    

41.    In an interference pattern formed by two coherent sources, the maximum and the minimum of the intensities are 9I₀ and I₀, respectively. The intensities of the individual waves are

    (a) 3I₀ and I₀

    (b) 4I₀ and I₀

    (c) 5I₀ and 4I₀

    (d) 9I₀ and I₀

Ans.    (b)

Sol.    

    Taking positive value, we get    

    a + b = 3a – 3b

    

42.    are two orthogonal states of a spin system. It is given that

    

    where represent the spin-up and spin-down states, respectively. When the system is in the state , its probability to be in spin-up state is ___________

Ans.    0.66

Sol.    

    According to Normalization condition,

    

    

    

43.    Neutrons moving with speed 10³ m/s are used for the determination of crystal structure. If the Bragg angle for the first order diffraction is 30º, the interplanar spacing of the crystal is ________Å.

    (Given: m_n = 1.675 × 10^–27 kg, h = 6.626 × 10^–34 Js)

Ans.    3.96

Sol.    Wavelength of neutrons can be given by m = 3.96 × 10^–10 m

    

    

44.    The Hamiltonian of a particle of mass 'm' is given by . Which of the following figures describes the motion of the particle in phase space?

    (a)

    (b)

    (c)

    (d)

Ans.    (d)

Sol.    For positive p arrow should point towards positive q and for negative p arrow should point towards negative q therefore correct answer is (d)

45.    The intensity of a laser in free space is 150 mW/m². The corresponding amplitude of the electric field of the laser is ______ V/m.        

Ans.    10.62

Sol.    

46.    The emission wavelength for the transition is 3122 Å. The ratio of populations of the final to the initial states at a temperature 5000 K is (h = 6.626 × 10^–34 J.s, c = 3 × 10⁸ m/s, k_B = 1.380 × 10^–23 J/K)

    (a) 2.03 × 10^–5

    (b) 4.02 × 10^–5

    (c) 7.02 × 10^–5

    (d) 9.83 × 10^–5

Ans.    (c)

Sol.    [Note: According to the option in the equation, the ration of proportion of the initial to the final state has to be calculate].

    The ratio of the molecule population of the initial state to final states is given by

    

    Here, difference in the energy states

    

    where, is the wavelength for the transition from state J = 2 to J = 3.

    

    

47.    Consider a system of 3 fermions, which can occupy any of the 4 available energy states with equal probability. The entropy of the system is

    (a) k_B ln 2

    (b) 2k_B ln 2

    (c) 2k_B ln 2

    (d) 3k_B ln 4

Ans.    (b)

Sol.    The distribution of fermions is

    

    So total number of microstates are 4, i.e., = 4.    

    The entropy of microstates are 4, i.e., = 4.    

48.    A particle is confined to a one dimensional potential box with potential

        

    If the particle is subjected to a perturbation, within the box, , where is a small constant, the first order correction to the ground state energy is

    (a) 0

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    For one dimension potential box, the wave function is given by

    

    

    First order correction to the ground state energy is

    

    

49.    Consider the process . The minimum kinetic energy of the muons (µ) in the centre of mass frame required to produce the pion pairs at rest is _________ MeV. (Given : m_µ = 105 MeV/c², = 140 MeV/c²)

Ans.    35

Sol.    According to the law of conservation of energy,

    (K.E. + rest mass energy) of reactance = (K.E. + rest mass energy) of products

    

    From equation (1),

    

50.    A one dimensional harmonic oscillator is in the superposition of number states, , given by

        

    The average energy of the oscillator in the given state is _____________ .

Ans.    3.25

Sol.    

    

51.    A nucleus X undergoes a first forbidden -decay to a nucleus Y. If the angular momentum (I) and parity (P), denoted by I^P as for X, which of the following is a possible I^P value for Y?

    (a)

    (b)

    (c)

    (d)

Ans.    (c)

Sol.    

    

    

52.    The current gain of the transistor in the following circuit is = 100. The value of collector current I_C is ______________ mA.

Ans.    1.5

Sol.     = 100

    I_C = ?

    V_BE = 0.7 V

    Apply DC analysis, it means capacitor is open circuit KVL at input

    KVL in input

    12 = (I_C + I_B)3 + 150 × I_B + 0.7 + 3 I_B                                    

    I_C = 100 I_B

    I_E = 101 I_B

    12 = 101 × I_B × 3 + 150 × I_B + 0.7 + 3 × 101 × I_B

    I_B = 0.015 mA

    I_C = .Ι_Β= 100 × 0.015 mA

    I_C = 1.5 mA    

53.    In order to measure a maximum of 1V with a resolution of 1mV using a n-bit A/D converter, working under the principle of ladder network, the minimum value of n is _____________

Ans.    10

Sol.    Given: Resolution = 1 mV

    To measure IV

    Resolutions

    Number of distinct values to measure 1V using least count of 1 mV

    

    number of bits required to code 1000 different amplitudes

    

54.    If L₊ and L_– are the angular momentum ladder operators, then, the expectation value of (L₊L_– + L_–L₊), in the state of an atom is ___________

Ans.    2

Sol.    

55.    A low pass filter is formed by a resistance R and a capacitance C. At the cut-off angular frequency , the voltage gain and the phase of the output voltage relative to the input voltage respectively, are

    (a) 0.71 and 45º

    (b) 0.71 and –45º

    (c) 0.5 and –90º

    (d) 0.5 and 90º    

Ans.    (b)

Sol.    LPF