## CSIR NET PHYSICS (2022)

Previous Year Question Paper with Solution.

1. The value of the integral , where m is a positive integer, is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Take m = 1

By substituting m = 1, option (c) is giving the correct option.

Let us check option,

2. At z = 0, the function of a complex variable z has

(a) No singularity

(b) A simple pole

(c) A pole of order 2

(d) A pole of order 3

Ans. (d)

Sol.

At z = 0, there is a pole of order 3.

3. Two n × n invertible real matrices A and B satisfy the relation

(AB)^{T} = –(A^{–1}B)^{–1}

If B is orthogonal then A must be

(a) Lower triangle

(b) Orthogonal

(c) Symmetric

(d) Anti-symmetric

Ans. (d)

Sol. (AB)^{T} = –(A^{–1}B)^{–1}

B^{T}A^{T} = –B^{–1}A

B^{–1}A^{T} = –B^{–1}A (Since B is orthogonal)

A^{T} = –A, A is asymmetric matrix.

4. The infinite series evaluated at x = , is

(a) 16

(b) 32

(c) 8

(d) 24

Ans. (a)

Sol.

diff. (1) w.r.t. x,

5. If z = (note that the exponent continues indefinitely), then a positive value of ln z is

(a) 2iln i

(b) ln i

(c) i ln i

(d) 2ln i

Ans. (b)

Sol. z = i^{Z}; lnz = zlog i

6. A wire, connected to a massless spring of spring constant k and a block of mass m, goes around of disc of radius a and moment of inertia I, as shown in the figure.

Assume that the spring remains horizontal, the pully rotates freely and there is no slippage between the wire and the pully. The angular frequency of oscillation of the disc is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Equation of motion

7. The Lagrangian of a system described by three generalized coordinates q_{1}, q_{2} and q_{3} is L = , where m, M and k are positive constants. Then, as a function of time

(a) Two coordinates remain constant and one evolves linearly

(b) One coordinates remain constant, one evolves linearly and third evolves as a quadratic function

(c) One evolves linearly and two evolves as a quadratic function

(d) Al three evolves linearly

Ans. (a)

Sol.

Which implies two coordinates remain constant and one evolves linearly.

8. The periods of oscillation of a simple pendulum at the sea level and at the top of a mountain of height 6 km are T_{1} and T_{2}, respectively. If the radius of earth is approximately 6000 km, then is closest to

(a) –10^{–4}

(b) –10^{–3}

(c) 10

(d) 10^{–3}

Ans. (d)

Sol.

9. A particle of rest mass m is moving with a velocity , with respect to an inertial frame S. The energy of the particle as measured by an observer S', who is moving with a uniform velocity with respect to S (in terms of and ) is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. v_{x} = 0; v_{y} = v; v_{z} = v; v = u

10. An electromagnetic wave is incident from vacuum normally on a planar surface of a non-magnetic medium. If the amplitude of the electric field of the incident wave is E_{0} and that of the transmitted wave is 2E_{0}/3, then neglecting any loss, the refractive index of the medium is

(a) 1.5

(b) 2.0

(c) 2.4

(d) 2.7

Ans. (b)

Sol.

11. A part of an infinitely long wire, carrying a current I, is bent in a semicircular arc of radius r (as shown in the figure).

The magnetic field at the centre O of the arc is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Magnetic field at o due to both line segments is zero.

Only magnetic field due to semi-circle

12. Two positive and two negative charges of magnitude q are placed on the alternate vertices of a cube of side a (as shown in the figure).

The electric dipole moment of this charge configuration is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

13. The electric and magnetic fields in an inertial frame are E = and B = , where a is a constant. A massive charged particle is released from rest. The necessary and sufficient condition that there is an inertial frame, where the trajectory of the particle is a uniform pitched helix, is

(a)

(b) –1 < a < 1

(c) a^{2} > 1

(d) –a^{2} > 2

Ans. (d)

Sol. The correct option is (d).

14. If the expectation value of the momentum of a particle in one dimension is zero, then its (box-normalizable) wavefunction may be of the form

(a) sin kx

(b) e^{ikx} sin kx

(c) e^{ikx} cos kx

(d) sin kx + e^{ikx} cos kx

Ans. (a)

Sol.

15. In terms of a complete set of orthonormal basis kets , n = 0, +1, +2, ..., the Hamiltonian is

where E and are constants. The state is an eigenstate with energy

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

16. The momentum space representing of the Schrodinger equation of a particle in a potential is , where and is a constant. The potential is (in the following V_{0} and a are constants)

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The correct option is (d).

17. Consider the Hamiltonian where A, B and C are positive constants, I is the 2 × 2 identity matrix and a Pauli matrices. If the normalized eigenvector corresponding to its largest energy eigenvalues is , then y is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

18. If the average energy of a quantum harmonic oscillator at a temperature T is such that = , then T satisfies

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For quantum Harmonic oscillator

It is given that

The energy of Harmonic oscillator can be written as

The partition function can be written as follows

19. A thermally isolated container, filled with an ideal gas at temperature T, is divided by a partition, which is clamped initially, as shown in the figure below.

The partition does not allow the gas in the two parts to mix. It is subsequently released and allowed to move freely with negligible friction . The final pressure at equilibrium is

(a) 5P/3

(b) 5P/4

(c) 3P/5

(d) 4P/5

Ans. (a)

Sol. PV = nRT

n = n_{1} + n_{2}

_{ }

20. A walker takes steps, each of length L, randomly in the directions along test, west, north and south. After four steps its distance from the starting point is d. The probability that d < 3L is

(a) 63/64

(b) 59/64

(c) 57/64

(d) 55/64

Ans. (d)

Sol. The correct option is (d).

21. An elastic rod has a low energy state of length L_{max} and high energy state of length L_{min}. The best schematic representation of the temperature (T) dependence of the mean equilibrium length L (T) of the rod, is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. The correct option is (d).

22. The circuit containing two n-channel MOSFETs shown below, works as

(a) a buffer

(b) a non-inverting amplifier

(c) an inverter

(d) a rectifier

Ans. (c)

Sol. The correct option is (c).

23. The figure below shows a circuit with two transistors, Q_{1} and Q_{2}, having current gains and respectively.

The collector voltage V_{c} will be closest to

(a) 0.9 V

(b) 2.2 V

(c) 2.9 V

(d) 4.2 V

Ans. (b)

Sol.

= 9.798 mA

V_{c} = 12 – 9.798 = 2.202 V

24. Four students (S_{1}, S_{2}, S_{3} and S_{4}) make multiple measurements on the length of a table. The binned data are plotted as histograms in the following figures

(a) S_{2}

(b) S_{1}

(c) S_{4}

(d) S_{3}

Ans. (b)

Sol. The correct option is (b).

25. A high impedence load network is connected in the circuit as shown below

The forward voltage drop for silicon diode is 0.7V and the Zener voltage is 9.10V. If the input voltage (V_{in}) is sine wave with an amplitude of 15V (as shown in the figure above), which of the following waveform qualitatively describes the output voltage (V_{out}) across the load?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The correct option is (c).

26. A bucket contains 6 red and 4 blue balls. A ball is taken out of the bucket at random and two balls of the same colour are put back. This step is repeated once more. The probability that the numbers of red and blue balls are equal at the end, is

(a) 4/11

(b) 2/11

(c) 1/4

(d) 3/4

Ans. (b)

Sol. The correct option is (b).

27. The value of the integral > 0, is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

z^{2} + 1 = 0

z = +i

Only z = +i will be located inside the semicircle.

28. The Laplace transform L[f](y) of the function f(x) = , n = 0, 1, 2, ... is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

29. The matrix corresponding to the differential operator in the space of polynomials of degree at most two, in the basis spanned by f_{1} = 1, f_{2} = x and f_{3} = x^{2}, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The correct option is (a).

30. The Lagrangian of a system of two particles . The normal frequencies are best approximated by

(a) 1.2 and 0.7

(b) 1.5 and 0.5

(c) 1.7 and 0.5

(d) 1.0 and 0.4

Ans. (d)

Sol.

31. The Lagrangian of a particle in one dimension is where a and V_{0} are positive constants. The best qualitative representation of a trajectory in the phase space is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The correct option is (b).

32. Earth may be assumed to be an axially symmetric freely rotating rigid body. The ratio of the principal moments of inertia about the axis of symmetry and an axis perpendicular to it is 33: 32. If T_{0} is the time taken by earth to make one rotation around its axis of symmetry, then the time period of precession is closest to

(a) 33 T_{0}

(b) 33 T_{0}/2

(c) 32 T_{0}

(d) 16 T_{0}

Ans. (c)

Sol. The correct option is (c).

33. A square conducting loop in the yz-plane, falls downward under gravity along the negative z-axis. Region 1, defined by z > 0 has a uniform magnetic field B = while region 2 (defined by z < 0) has no magnetic field.

The time dependence of the speed v(t) of the loop, as it starts to fall from well within the region 1 and passes into the region 2, is best represented by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The correct option is (b).

34. Two small metallic objects are embedded in a weakly conducting medium of a conducting and dielectric constant . A battery connected between them leads to a potential difference V_{0}. It is subsequently disconnected at time t = 0. The potential difference at a later time t is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

35. A stationary magnetic dipole m = is placed above an infinite surface (z = 0) carrying a uniform surface current density . The torque on the dipole is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The correct option is (a).

36. Two parallel conducting rings, both of radius R, are separated by a distance R. The planes of the rings are perpendicular to the line joining their centres, which is taken to the x-axis.

If both the rings carry the same current i along the same direction, the magnitude of the magnetic field along the x-axis is best represented by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We know that the magnetic field on the axis of circular loop at a distance x from center

The field will be maximum at the center of loop.

Now there are two loops. So, the magnetic field in the region in between two loops will be

, The field will be maximum (Helmotz Coil) in the region in between two loops. However, there will be decay of filed on either side of loops. So, the correct option is (a).

37. At time t = 0, a particle is in the ground state of the Hamiltonian H(t) = where and m are positive constants. To , the probability that at t = , the particle would be in the first excited state of H(t = 0) is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

38. To first order in perturbation theory, the energy of the ground state of the Hamiltonian

(treating the third term of the Hamiltonian as a perturbation) is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

39. The energy/energies E of the bound state(s) of a particle of mass m in one dimension in the potential (where V_{0} > 0) is/are determined by

(a)

(b)

(c)

(d)

Ans. (c)

Sol. V = +ve

E = –ve

E < –V

40. The energy levels of a system, which is in equilibrium at temperature are 0, a 2. If two identical bosons occupy these energy levels, the probability of the total energy being 3, is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Partition function

The probability of the total energy being 3

41. A paramagnetic salt with magnetic moment ion (where µ_{B} is the Bohr magneton) is in the thermal equilibrium at temperature T in a constant magnetic field B. The average magnetic moment , as a function of , is best represented by

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The partition function can be written as follows

So, the correct graph is represented in option (c).

42. A system of N non-interacting particles in one-dimension, each of which is in a potential V(x) = gx^{6} where g > 0 is a constant and x denotes the displacement of the particle from its equilibrium position. In thermal equilibrium, the heat capacity at constant volume is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Using virial theorem

43. A liquid oxygen cylinder system is fitted with a level-sensor (L) and a pressure-sensor (P), as shown in the figure below. The outputs of L and P are set to logic high (S = 1) when the measured values exceed the respective preset threshold values. The system can be shut off either by an operator by setting the input S to high, or when the level of oxygen in the tank falls below the threshold value.

The logic gates X, Y and Z, respectively are

(a) OR, AND and NOT OR

(b) AND, OR and NOT

(c) NAND, OR and NOT

(d) NOR, AND and NOT

Ans. (b)

Sol. The correct option is (b).

44. A high frequency voltage signal V_{i} = V_{m}sin is applied to a parallel plate deflector as shown in the figure

An electron beam is passing through the deflector along the central line. The best qualitative representation of the intensity I(t) of the beam after it goes through the narrow circular aperture D, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

At zero applied voltage there will be no deflection of charge particle. As a result, we will get I. But, in other value of , the applied voltage is non zero. Thus, there will be deflection of charge particle. So, it can not reach D. Thus I(t) will be zero.

45. An amplifier with a voltage gain of 40 dB without feedback is used in an electronic circuit. A negative feedback with a fraction 1/40 is connected to the input of this amplifier. The net gain of the amplifier in the circuit is closest to

(a) 40 dB

(b) 37 dB

(c) 29 dB

(d) 20 dB

Ans. (c)

Sol.

In negative feedback

Gain in dB 20 log 28.9 dB = 29 dB

46. A receiver operating at 27°c has an input resistance of . The input thermal noise voltage for this receiver with a bandwidth of 100kHz is closest to

(a) 0.4 nV

(b) 0.6 pV

(c) 40 mV

(d) 0.4 µV

Ans. (d)

Sol.

47. The Raman rotational-vibrational spectrum of nitrogen molecules is observed using an incident radiation of wavenumber 12500 cm^{–1}. In the first shifted band, the wavenumbers of the observed lines (in cm^{–1}) are 10150, 10158, 10170, 10182 and 10190. The values of vibrational frequency and rotational constant (in cm^{–1}), respectively are

(a) 2330 and 2

(b) 2350 and 2

(c) 2350 and 3

(d) 2330 and 3

Ans. (a)

Sol. Central one will be the corresponding value of vibrational frequency 10150, 10158, 10170, 10182 and 10190. Here 10170 in the Strokes line corresponding to vibrational one.

= 12500 cm^{–1}

_{Stroke} = 10170 cm^{–1}

= 12500 – 10170 = 2330 cm^{–1}

10150 and 10158 are rotational Raman Strokes lines. Also, 10182 and 10190 are rotational Raman Strokes lines.

We know, 1058 – 1050 = 8 cm^{–1} = 4B B = 2cm^{–1}

48. The electronic configuration of ^{12}C is 1s^{2}2s^{2}2p^{2}. Including LS coupling, the correct ordering of its energies is

(a) E(^{3}P_{2}) < E(^{3}P_{1}) < E(^{3}P_{0}) < E(^{1}D_{2})

(b) E(^{3}P_{0}) < E(^{3}P_{1}) < E(^{3}P_{2}) < E(^{1}D_{2})

(c) E(^{1}D_{2}) < E(^{3}P_{2}) < E(^{3}P_{1}) < E(^{3}P_{0})

(d) E(^{3}P_{1}) < E(^{3}P_{0}) < E(^{3}P_{2}) < E(^{1}D_{2})

Ans. (b)

Sol. 1s^{2}2s^{2}2p^{2}

^{ }

^{ }The lower energy belong from higher S(=1), less than half field (2p^{2})

S = 1, L = 1, J = |L + S|.....|L – S| = 2, 1, 0 Term = ^{3}P_{0, 1, 2}

S = 0, L = 0, J = |L + S|.....|L – S| = 0 Term = ^{1}S_{0}

S = 0, L = 2, J = |L + S|.....|L – S| = 2 Term = ^{1}D_{2}

According to Hund's rule, Higher S, lower energy. Higher L, lower will be energy and Higher J, higher will be the energy.

By applying Hund's rule E(^{3}P_{0}) < E(^{3}P_{1}) < E(^{3}P_{2}) < E(^{1}D_{2}).

49. In the absorption spectrum of H-atom, the frequency of transition from the ground state to the first excited state is v_{H}. The corresponding frequency for a bound state of a positively charged muon (µ^{+}) and an electron is v_{µ}. Using m_{µ} = 10^{–2} kg, m_{e} = 10^{–30} kg and m_{p} >> m_{e}, m_{µ}, the value of (v_{µ} – v_{H})/v_{H} is

(a) 0.001

(b) –0.001

(c) –0.01

(d) 0.01

Ans. (c)

Sol.

For muonium

50. The energies of a two-level system are +E. Consider an ensemble of such non-interacting systems at a temperature T. At low temperatures, the leading term in the specific heat depends on T as

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The partition function can be written as follows

51. The figures (i), (ii) and (iii) below represent an equilateral triangle, a rectangle and a regular hexagon, respectively.

Which of these can be primitive unit cells of a Bravais lattice in two dimensions?

(a) only (i) and (iii) but not (ii)

(b) only (i) and (ii) but not (iii)

(c) only (ii) and (iii) but not (i)

(d) all of them

Ans. (c)

Sol. The correct option is (c).

52. The Hamiltonian for a spin –1/2 particle in a magnetic field B = is given by H = S.B, where S is its spin (in units of ) and is a constant. If the average spin density is for an ensemble of such non-interacting particles, then

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

53. The tensor component of the nuclear force may be inferred from the fact that deuteron nucleus

(a) has only one bound state with total spin S = 1

(b) has a non-zero electric quadrupole moment in its ground state

(c) is stable while triton is unstable

(d) is the only two nucleon bound state

Ans. (b)

Sol. The tensor component of the nuclear force may be inferred from the fact has a non-zero electric quadrupole moment in its ground state.

54. The elastic scattering process may be treated as a hard-sphere scattering. The mass of , where m_{p} 938MeV/c^{2} is the mass of proton. The total scattering cross-section is closest to

(a) 0.01 milli barn

(b) 1 milli-barn

(c) 0.1 barn

(d) 10 barn

Ans. (c)

Sol.

55. Thermal neutrons may be detected most efficiently by a

(a) Li^{6} loaded plastic scintillator

(b) Geiger-Muller counter

(c) inorganic scintillator CaF2

(d) silicon detector

Ans. (a)

Sol. Thermal neutrons may be detected most efficiently by 6 Li loaded plastic scintillator.