1. A horizontal metal disc rotates about the vertical axis in a uniform magnetic field pointing up as shown in the figure. A circuit is made by connecting one end A of a resistor to the centre of the disc and the other end B to its edge through a sliding contact S. The current that flows through the resistor is

(a) zero

(b) DC from A to B

(c) DC from B to A

(d) AC

Ans. (c)

Sol. Due to rotation force on electrons (negative charge), (towards axis of rotation)

Hence, centre is negatively charged and edge is positvely charged.

So, current will flow from B to A.

2. A particle is in the state in the eigenbasis of S² and S_z. If we measure S_z the probabilities of getting and , respectively, are

(a)

(b)

(c) 0 and 1

(d)

Ans. (b)

Sol. Given : A spin-½ particle is in the state

Measurement of S_Z gives and with respective eigenbasis and

Method-1:

Representing

Probability of finding S_Z as and are and respectively.

Method-2:

Initial state of particle

Final state of finding S_Z as are and respectively.

3. Which of the following functions cannot be the real part of a complex analytic function of z = x + iy?

(a) x² y

(b) x² – y²

(c) x³ – 3xy²

(d) 3x²y – y – y³

Ans. (a)

Sol. The real part and imaginary part of a complex analytic function of z = x + iy, will be harmonic in nature i.e. both will be satisfy Laplace's equation. Suppose, 'u' be the real part, then

4. The motion of a particle of mass m in one dimension is described by the Hamiltonian . What is the difference between the (quantized) energies of the first two levels? (In the following, is the expectation value of x in the ground state)

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given : Hamiltonian of the system :

First order correction to energy of the n^th state.

Second order correction to energy of the n^th state.

The non-zero contribution comes only from m = (n + 1) and (n – 1)

Energy difference between two levels is

5. Let denote the eigenfunctions of a Hamiltonian for a spherically symmetric potential V(r). The expectation value of L_z in the state is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

6. Three identical spin fermions are to be distributed in two non-degenerate distinct energy levels. The number of ways this can be done is

(a) 8

(b) 4

(c) 3

(d) 2

Ans. (a)

Sol.

But in present case,

= (2S + 1)^N

7. Let A, B and C be functions of phase space variables (coordinates and momenta of a mechanical system). If {,} represents the Poisson bracket, the value of {A, {B, C}} – {{A, B}, C} is given by

(a) 0

(b) , A}}

(c) {A, {C, B}}

(d) {{C, A}, B}

Ans. (d)

Sol.

Jacobi identity

8. If A, B and C are non-zero Hermitian operators, which of the following relations must be false?

(a) [A, B] = C

(b) AB + BA = C

(c) ABA = C

(d) A + B = C

Ans. (a)

Sol.

So, C = [A, B] is not hermitian.

9. The expression

is proportional to

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

and for others we can also calculate

10. Given that the integral , the value of is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

11. The force between two long and parallel wires carrying currents I₁ and I₂ and separated by a distance D is proportional to

(a) I₁I₂/D

(b) (I₁ + I₂)/D

(c) (I₁ I₂/D)²

(d) I₁ I₂ / D²

Ans. (a)

Sol. Force per unit length on each wire is given by

12. A loaded dice has the probabilities and of turning up 1, 2, 3, 4, 5 and 6, respectively. If it is thrown twice, what is the probability that the sum of the numbers that turn up is even?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. If the loaded dice is thrown twice, the sum of the numbers that turnup will be even if both number will be even or both will be odd.

Required probability = P(Even – Even) + P(Odd – Odd)

13. A particle moves in a potential . Which component (s) of the angular momentum is / are constant (s) of motion?

(a) none

(b) L_x, L_y and L_z

(c) only L_x and L_y

(d) only L_z

Ans. (d)

Sol.

In spherical polar coordinates,

Lagrangian of system is

L = T – V

Therefore is cyclic coordinater then we can say that p_Φ is constant.

is equal to L_z. Therefore L_z is constant of motion.

14. The Hamiltonian of a relativistic particle of rest mass m and momentum p is given by , in units in which the speed of light c = 1. The corresponding Lagrangian is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

This is Hamiltonian of a relativistic particle with c = 1. Its very well known that Lagrangian of a relativistic particle is

Taking , and c = 1, we get

15. A ring of mass m and radius R rolls (without slipping) down an inclined plane starting from rest. If the centre of the ring is initially at a height h, the angular velocity when the ring reaches the base is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Let = speed of the ring when it reaches the base.

Therefore, total K.E. of the ring

Height descended by the centre of the ring = h – R

Therefore, loss in P.E. = gain in K.E.

mg (h – R) = mv²

Angular velocity

16. Consider the op-amp circuit shown in the figure.

If the input is a sinusoidal wave V_i = 5 sin(1000t), then the amplitude of the output V₀ is

(a)

(b) 5

(c)

(d)

Ans. (c)

Sol.

17. If one of the inputs of a J-K flip flop is high and the other is low, then the outputs Q and

(a) oscillate between low and high in race-around condition

(b) toggle and the circuit acts like a T flip flop

(c) are opposite to the inputs

(d) follow the inputs and the circuit acts like an R-S flip flop

Ans. (d)

Sol. J-K flip-flop follow the I/P and the circuit acts like an R-S flip flop

Through table of JK flip-flop

Truth table of RS flip-flop

18. Two monochromatic sources, L₁, and L₂, emit light at 600 and 700 nm, respectively. If their frequency bandwidths are 10^–1 and 10^–3 GHz, respectively, then the ratio of linewidth of L₁ and L₂ is approximately

(a) 100 : 1

(b) 1 : 85

(c) 75 : 1

(d) 1 : 75

Ans. (c)

Sol.

19. Let (V, A) and (V', A') denote two sets of scalar and vector potentials, and a scalar function. Which of the following transformations leave the electric and magnetic fields (and hence Maxwell's equations) unchanged?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We can choose different sets of so that fields remains unchanged after transformation.

20. Consider the melting transition of ice into water at constant pressure. Which of the following thermodynamic quantities does not exhibit a discontinuous change across the phase transition?

(a) internal energy

(b) Helmholtz free energy

(c) Gibbs free energy

(d) entropy

Ans. (c)

Sol. Ice to water at constant pressure is first order phase transition.

In first order phase transition, g is continuous but first order derivative of g is discontinuous.

21. Two different thermodynamic systems are described by the following equations of state:

and where T^(1,2), N^(1,2) are respectively, the temperatures, the mole numbers and the internal energies of the two systems, and R is the gas constant. Let U_tot denote the total energy when these two systems are put in contact and attain thermal equilibrium. The ratio is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Let and be the temperature and internal energies of first and second system respectively, when both systems are in thermal equilibrium.

When both of the systems are in thermal contact then energy exchange takes place as

When both of the systems attain thermal equilibrium their temperatures become equal.

Then,

this must be the correct answer.

If U⁽¹⁾ is the energy of first system in thermal equilibrium then

22. The molecules of mass m of an ideal gas obeys Maxwell's velocity distribution law at an equilibrium temperature T. Let (v_x, v_y, v_z) denote the components of the velocity and k_B the Boltzmann constant. The average value of , where and are constants, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

23. The entropy S of a themodynamic system as a function of energy E is givne by the following graph

The temperatures of the phases A, B and C, denoted by T_A, T_B and T_C, respectively, satisfy the following inequalities:

(a) T_C > T_B > T_A

(b) T_A > T_C > T_B

(c) T_B > T_C > T_A

(d) T_B > T_A > T_C

Ans. (c)

Sol.

24. The physical phenomenon that cannot be used for memory storage applications is

(a) large variation in magnetoresistance as a function of applied magnetic field

(b) variation in magnetization of a ferromagnet as a function of applied magnetic field

(c) variation in polarization of a ferroelectric as a function of applied electric field

(d) variation in resistance of a metal as a function of applied electrid field

Ans. (d)

Sol. Variation in resistance of metal as a function of electric field cannot store the memory.

25. Two identical Zener diodes are placed back to back in series and are connected to a variable DC power supply. The best representation of the I-V characteristics of the circuit is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The best representation of the I-V characteristics of the circuit is (with DC supply)

26. A pendulum consists of a ring of mass M and radius R suspended by a massless rigid rod of length l attached to its rim. When the pendulum oscillates in the plane of the ring, the time period of oscillation is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Time period for compound pendulum

27. Spherical particles of a given material of density  are released from rest inside a liquid medium of lower density. The viscous drag force may be approximated by the Stoke's law, i.e., , where is the viscosity of the medium, R the radius of a particle and v its instantaneous velocity. If is the time taken by a particle of mass m to reach half its terminal velocity, then the ratio is

(a) 8

(b) 1/8

(c) 4

(d) 1/4

Ans. (c)

Sol. Drag force on particles ... (1)

Mass of a particle ... (2)

Eq. (2) put in Eq. (1)

Eq. (4), (3) put in Eq. (5)

when terminal velocity is reached, ... (7)

Eq. (7) put in Eq. (6)

Now, if be the time to reach half the terminal velocity then from (6)

Eq. (8) put in Eq. (9)

28. A system of N classical non-interacting particles, each of mass m, is at a temperature T and is confined by the external potential (where A is a constant) in three dimensions. The internal energy of the system is

(a) 3Nk_BT

(b)

(c) N(2mA)^3/2k_BT

(d)

Ans. (a)

Sol. The contribution of thermal energy with each quadratic term in hamiltonian is .

The hamiltonian of a classical particle

IN 3-D, hamiltonian has 6 quadratic term. So, the internal energy of a particle is

The internal of a system of N particles = 3 NkT.

29. Consider a particle of mass m attached to two identical springs each of length l and spring constant k (see the figure below). The equilibrium configuration is the one where the springs are unstretched. There are no other external forces on the system. If the particle is given a small displacement along the x-axis, which of the following describes the equation of motion for small oscillations?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Let x be the displacement of particle along x-axis.

Net force on the particle along 'x' axis = –2F cos

Where, F = k × elongation

Therefore, using Newton's law we get

Net force

Using binomial expansion {keep only upto x² term}

We get,

30. If is the eigenfunction of a one dimensional Hamiltonian with eigenvalue E = 0, the potential V(x) (in units where ) is

(a) 12x²

(b) 16x⁶

(c) 16x⁶ + 12x²

(d) 16x⁶ – 12x²

Ans. (d)

Sol. Given : is an eigenfunction of a 1-D hamiltonian with eigenvalue E = 0

From 1-D Schrodinger equation,

31. The electric field of an electromagnetic wave is given by . The associated magnetic field is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

32. The energy of an electron in a band as a function of its wave vector k is given by E(k) = E₀ – B(cos k_xa + cos k_ya + cos k_za), where E₀, B and a are constants. The effective mass of the electron near the bottom of the band is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. E(k) = E₀ – B (cos k_xa + cos k_ya + cos k_za)

Near the bottom of the band (using Taylor series expansion)

33. A DC voltage V is applied across a Josephson junction between two superconductors with a phase difference . If I₀ and k are constants that depend on the properties of the junction, the current flowing through it has the form

(a)

(b)

(c)

(d)

Ans. (a)

Sol. On the application of DC voltage. AC current flows across the junction and that is given by

34. Consider the following ratios of the partial decay widths and . If the effects of electromagnetic and weak interactions are neglected, then R₁ and R₂ are, respectively,

(a) 1 and

(b) 1 and 2

(c) 2 and 1

(d) 1 and 1

Ans. (d)

Sol. First Method :

The reaction cross-sectional are directly proportional to the Clebsch-Gordan coefficients.

Consider I = j, I₃ = m

Therefore, j = 1, j₁ = 1, j₂ = 1

m₁ = +1, m₂ = 0, m = +1

C.G. coefficient

j = 1, m = –1, j₁ = 1, j₂ = 1, m₁ = –1, m₂ = 0

R₂ = 1:1 = 1

Therefore, R₁ : R₂ = 1:1

Second Method :

The decay widths (where is the life time of the particles)

35. The intrinsic electric dipole moment of a nucleus

(a) increases with Z, but independent of A

(b) decreases with Z, but independent of A

(c) is always zero

(d) increases with Z and A

Ans. (c)

Sol. The electric dipole moments of the nucleus is non-zero only if the total charge of the nucleus is stufted to one side of the nucleus. Else, the nuclei do not have any non-zeroelectric dipole moment due to their definite parity.

36. According to the shell model, the total angular momentum (in units of ) and the parity of the ground state of the nucleus is

(a) with negative parity

(b) with positive parity

(c) with positive parity

(d) with negative parity

Ans. (a)

Sol.

Number of protons

Number of neutrons N = 4 (even) do not contribute any role.

37. A point charge q is placed symmetrically at a distance d from two perpendicularly placed grounded conducting infinite plates as shown in the figure. The net force on the charge (in units of ) is

(a) away from the corner

(b) towards the corner

(c) towards the corner

(d) away from the corner

Ans. (b)

Sol.

38. Let four point charges q, –q/2, q and –q/2 be placed at the vertices of a square of side a. let another point charge –q be placed at the centre of the square (see the figure).

Let V(r) be the electrostatic potential at a point P at a distance r >> a from the centre of the square. Then V(2r)/V(r) is

(a) 1

(b)

(c)

(d)

Ans. (d)

Sol.

39. Let A and B be two vectors in three-dimensional Euclidean space. Under rotation, the tensor product T_ij = A_iB_j

(a) Reduces to a direct sum of three 3-dimensional representations

(b) is an irreducible 9-dimensional representation

(c) reduces to a direct sum of a 1-dimensional, a 3-dimensional and a 5-dimensional irreducible representations

(d) reduces to a direct sum of a 1-dimensional and an 8-dimensional irreducible representation

Ans. (c)

Sol. The rank-2 tensor T_ij = A_iB_j can be expressed as a 3 × 3 matrix.

T_ij can be decomposed into irreducible representation in the following way:

Here, is rank 0 tensor and transform under rotation like a scalar and it will have only one independent component.

Here, is a rank 1 tensor and transform under rotation like a vector and represented by

i.e. there are three independent component.

Here, is a rank 2 tensor and represented by

Since, is symmetric and traceless, there are 5 independent elements.

40. Fourier transform of the derivative of the Dirac -function, namely , is proportional to

(a) 0

(b) 1

(c) sink

(d) ik

Ans. (d)

Sol. Fourier transform of

Using the property of Dirac delta function

So,

41. A particle is in the ground state of an infinite square well potential given by,

The probability to find the particle in the interval between and is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given :

Symmetrical infinite potential well of width '2a'

So, wave function of the particle in ground state,

Probability of finding the particle in the interval between to is

42. The expectation value of the x-component of the orbital angular momentum L_x in the state (where are the eigenfunctions in usual notation), is

(a)

(b) 0

(c)

(d)

Ans. (a)

Sol.

43. A particle is prepared in a simultaneous eigenstate of L² and L_z. If and are respectively the eigenvalues of L² and L_z, then the expectation value of the particle in this state satisfies

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

For a given l, m varies from –l to l.

44. If the electrostatic potential in a charge free region has the form , then the functional form of f(r) (in the following a and b are constants) is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

According Laplace equation is

The value of f(r) in option (b) satisfy the equation (i).

45. If , then the integral (where C is along the perimeter of a rectangular area bounded by x = 0, x = a and y = 0, y = b) is

(a)

(b)

(c)

(d) 0

Ans. (d)

Sol.

Since, line integral of a conservative field along a closed path is zero, then

46. Consider an n × n(n > 1) matrix A, in which A_ij is the product of the indices i and j (namely A_ij = ij). The matrix A

(a) has one degenerate eigenvalue with degeneracy (n –1)

(b) has two degenerate eigenvalues with degeneracies 2 and (n –2)

(c) has one degenerate eigenvalue with degeneracy n

(d) does not have any degenerate eigenvalue

Ans. (a)

Sol. Given : A_ij = ij. So, for a 3 × 3 matrix

Since, the second row and third row is multiple of first row and the same is true for columns, then eigenvalue of the matrix will be Tr(A), 0, 0 i.e. 14, 0, 0

'0' is a degenerate eigenvalue with degeneracy '2'

In general, matrix A (order 'n') has a degenerate eigenvalue with degeneracy 'n – 1'.

47. A child makes a random walk on a square lattice of lattice constant a taking a step in the north, east, south, or west directions with probabilities 0.255, 0.255, 0.245, and 0.245, respectively. After a large number of steps, N, the expected position of the child with respect to the starting point is at a distance

(a) in the north-east direction

(b) in the north-east direction

(c) in the south-east direction

(d) 0

Ans. (a)

Sol. Single step length = a

After a single step, the expected value of position of the child.

Expected position (NORTH-EAST DIRECTION)

48. A carnot cycle operates as a heat engine between two bodies of equal heat capacity until their temperatures become equal. If the initial temperatures of the bodies are T₁ and T₂, respectively, and T₁ > T₂ then their common final temperature is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Carnot cycle is reversible cycle so that the entropy change must be zero i.e. = 0

Let the equilibrium temperature be T then

49. Three sets of data A, B and C from an experiment, represented by ×, and , are plotted on a log-log scale. Each of these are fitted with straight lines as shown in the figure.

The functional dependence y(x) for the sets A, B and C are, respectively

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Since, datas are plotted on log-log scale and all passes through (1, 1) then the straight lines will be of the nature, log y = m log x

50. A sample of Si has electron and hole mobilities of 0.13 and 0.05 m²/V-s respectively at 300K. It is doped with P and Al with doping densities of 1.5 × 10²¹/m³ and 2.5 × 10²¹/m³ respectively. The conductivity of the doped Si sample at 300K is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Si doped with P and Al.

As doping of Al > doping of P.

Hence, semiconductor will behave as p-type semiconductor.

Majority carrier concentration (2.5 × 10²¹ – 1.5 × 10²¹) = 1 × 10²¹m³

51. A 4-variable switching function is given by , where d is the do-not-care-condition. The minimized form of f in sum of products (SOP) form is

(a)

(b)

(c) AD + BC

(d)

Ans. (d)

Sol. 4-variable switching function is given by

There are two Quad.

52. A perturbation V_pert = aL² is added to the Hydrogen atom potential. The shift in the energy level of the 2P state, when the effects of spin are neglecte up to second order in a, is

(a) 0

(b)

(c)

(d)

Ans. (c)

Sol. Method-1:

The energy corrected to second order

E = E⁽⁰⁾ + E⁽¹⁾ + E⁽²⁾

The shift in energy = E – E⁰ = E⁽¹⁾ + E⁽²⁾

where, E⁽⁰⁾ = unperturbed energy

E⁽¹⁾ = first order correction in energy

E⁽²⁾ = second order correction in energy

Perturbation, V_pert = aL²

2p state of H-atom is 3-fold degenerate. Secular determinant to find first order correction in energy.

The second order correction in energy for degenerate perturbation theory includes matrix element (that has summation over non-deg states). There is no non-deg states, so E⁽²⁾ = 0

The shift in energy =

Methode-2:

Since the energy correction to the alldegenerate states are equal. Therefore, we can use the non-degenerate perturbation theory to solve the problem as follows.

Given : is added to a hydrogen atom potential.

First order correction to energy of the 2p state

Second order correction to energy of the 2p state

Therefore, the shift in the energy level

53. A gas laser cavity has been designed to operate at = 0.5µm with a cavity length of 1m. With this set-up, the frequency is found to be larger than the desired frequency by 100 Hz. The change in the effective length of the cavity required to returne the laser is

(a) –0.334 × 10^–12 m

(b) 0.334 × 10^–12 m

(c) 0.167 × 10^–12 m

(d) –0.167 × 10^–12 m

Ans. (d)

Sol. The separation between cavity mirros, where 'm' is mode number.

Charge in length of cavity,

54. The spectroscopic symbol for the ground state of ₁₃Al is ²P_1/2. Under the action of a strong magnetic field (when L-S coupling can be neglected) the ground state energy level will split into

(a) 3 levels

(b) 4 levels

(c) 5 levels

(d) 6 levels

Ans. (d)

Sol.

Out of the given 6 states, 2 states have same energy shift due to strong magnetic field. Therefore, the ground state energy level will split into five levels.

55. A uniform linear monotomic chain is modeled by a spring-mass system of masses m separated by nearest neighbor distance a and spring constant . The dispersion relation for this system is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Frequency of monoatomic one dimensional lattice is given by

where 'c' is the force constant (spring constant)