CSIR NET PHYSICS (DEC-2019)
Previous Year Question Paper with Solution.

1. The angular frequency of oscillation of a quantum harmonic oscillator in two dimensions is If it is in contact with an external heat bath at temperature T, its partition function is (in the following )

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

2. A student measures the displacement x from the equilibrium of a stretched spring and reports it be 100 µm with a 1% error. The spring constant k is known to be 10 N/m with 0.5% error. The percentage error in the estimate of the potential energy is

(a) 0.8%

(b) 2.5%

(c) 1.5%

(d) 3.0%

Ans. (b)

Sol. Percentage error in potential energy

3. The Hamiltonian of two interacting particles one with spin 1 and the other with spin is given by , where and denote the spin operators of the first and second particles, respectively and A and B are positive constants. The largest eigenvalue of this Hamiltonian is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

4. Consider the set of polynomials in t of degree less than n, such that x(0) = 0 and x(1) = 1. This set

(a) constitutes a vector space of dimension n

(b) constitutes a vector space of dimension n – 1

(c) constitutes a vector space of dimension n – 2

(d) does not constitute a vector space

Ans. (d)

Sol. x(t) = a0 + a1t + a2t2 + ... + an – 1yn – 1

x(0) = 0

0 = a0

x(t) = a1t + a2t2 + ... + an – 1tn – 1

also, x(1) = 1

1 = a1 + a2 + .... + an–1 ... (i)

t, t2, t3, .... will make basis vector if

c1t + c2t2c3t3 + ..... = 0

If c1 = c2 = c3 = .... = 0

Which is contradicting with (i)

So, It does not constitute a vector space. For our case if a1 = a2 = .... = 0

Its summation can't be = 1.

5. Consider black body radiation in thermal equilibrium contained in a two-dimensional box. The dependence of the energy density on the temperature T is

(a) T3

(b) T

(c) T2

(d) T4

Ans. (a)

Sol. The energy density at the temperature T is,

Energy density for 2D photon, Riemann Zeta function

6. The energy eigenvalues of a particle of mass m , confined to a rigid one-dimensional box of width L, are En(n = 1, 2,....). If the walls of the box are moved very slowly toward each other, the rate of change of time-dependent energy of the first excited state is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

7. A ball, initially at rest, is dropped from a height h above the floor bounces again and again vertically. If the coefficient of restitution between the ball and the floor is 0.5 , the total distance traveled by the ball before it comes to rest is

(a)

(b)

(c) 3h

(d) 2h

Ans. (b)

Sol.

8. Two spin fermions of mass m are confined to move in a one-dimensional infinite potential well of width L. If the particles are known to be in a spin triplet state, the ground state energy of the system (in units of ) is

(a) 8

(b) 2

(c) 3

(d) 5

Ans. (d)

Sol. If probability in triplet state means . So one electron in n = 1 state and another in n = 2 state. So ground state energy of configuration is

9. The figure below shows a 2-bit simultaneous analog-to-digital (A/D) converter operating in the voltage range 0 to V0. The output of the comparators are C1, C2 and C3 with the reference inputs V0/4, V0/2 and 3V0/4, respectively. The logic expression for the output corresponding to the less significant bit is

(a) C1C2C3

(b)

(c)

(d)

Ans. (c)

Sol. Least significant bit is (0, 1) i.e. C1 will be selected and C2 = 0, C3 = 0

10. The yz-plane at x = 0 carries a uniform surface charge density . A unit point charge is moved from a point (, 0, 0) on one side of the plane to a point (–, 0, 0) on the other side. If is an infinitesimally small positive number, the work done in moving the charge is

(a) 0

(b)

(c)

(d)

Ans. (a)

Sol. Work done = q[V(b) – V(a)]

Potential is same so that w = 0

11. A circular conducting wire loop is placed close to a solenoid as shown in the figure below. Also shown is the current through the solenoid as a function of time.

The magnitude |i(t)| of the induced current in the wire loop, as a function of time t , is best represented as

(a)

(b)

(c)

(d)

Ans. (d)

Sol. According to Faraday's law

So when current increases, |I(t)| will increase and when it will decrease |I(t)| will decrease.

12. A mole of gas at initial temperature Ti comes into contact with a heat reservoir at temperature Tf and the system is allowed to reach equilibrium at constant volume. If the specific heat of the gas is CV = T, where is a constant, the total change in entropy is

(a) zero

(b)

(c)

(d)

Ans. (d)

Sol. Change in entropy of gas

Tds = CVdT + PdV, dV = 0

Chang in entropy of reservoir (at constant temperature Tf)

13. An ideal Carnot engine extracts 100 J from a heat source and dumps 40 J to a heat sink at 300 K. The temperature of the heat source is

(a) 600 K

(b) 700 K

(c) 750 K

(d) 650K

Ans. (c)

Sol. Q1 = 100 J, Q2 = 40 J

T1 = ? T2 = 300 K

14. A block of mass m, attached to a spring, oscillates horizontally on a surface. The coefficient of friction between the block and the surface is µ. Which of the following trajectories best describes the motion of the block in the phase space ( xpx-plane)?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Due to friction amplitude and momentum of oscillation continuously decreases.

15. Let C be the circle of radius centered at in the complex z-plane that is traversed counter-clockwise. The value of the contour integral is

(a) 0

(b)

(c)

(d)

Ans. (c)

Sol.

16. If the rank of an n × n matrix A is m, where m and n are positive integers with 1< m < n, then the rank of the matrix A2 is

(a) m

(b) m – 1

(c) 2m

(d) m – 2

Ans. (a)

Sol.

(b), (c), (d) can't be correct so option (a) is correct.

17. A particle of mass m is confined to a box of unit length in one dimension. It is described by the wavefunction for 0 < x < 1 and zero outside this interval. The expectation value of energy in this state is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

18. In the circuit below, D is an ideal diode, the source voltage VS = V0 sin is a unit amplitude sine wave and RS = RL

The average output voltage, VL, across the load resistor RL is

(a)

(b)

(c) 3V0

(d) V0

Ans. (a)

Sol.

19. The normalized wavefunction of a particle in three dimensions is given by

where a is a positive constant and N is a normalization constant. If L is the angular momentum operator, the eigenvalues of L2 and Lz, respectively, are

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

20. The electric field of an electromagnetic wave is . The average flow of energy per unit area per unit time, due to this wave, is

(a) 27 × 104 W/m2

(b) 27 × 10–4 W/m2

(c) 27 × 10–2 W/m2

(d) 27 × 102 W/m2

Ans. (b)

Sol.

21. The energies available to a three state system are 0, E and 2E , where E > 0. Which of the following graphs best represents the temperature dependence of the specific heat?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Correct option is (d)

22. The values of a and b for which the force is conservative are

(a) a = 2, b = 3

(b) a = 1, b = 3

(c) a = 2, b = 6

(d) a = 3, b = 2

Ans. (a)

Sol.

23. A positively charged particle is placed at the origin (with zero initial velocity) in the presence of a constant electric and a constant magnetic field along the positive z and x-directions, respectively. At large times, the overall motion of the particle is a drift along the

(a) positive y-direction

(b) negative z-direction

(c) positive z-direction

(d) negative y-direction

Ans. (a)

Sol. Initially charged particle will experience electric force and will gain velocity then it will deflect in magnetic field

24. A box contains 5 white and 4 black balls. Two balls are picked together at random from the box. What is the probability that these two balls are of different colours?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Probability that the two balls are of different colors

5W, 4B

25. Which of the following terms, when added to the Lagrangian of a system with two degrees of freedom will not change the equations of motion?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

26. The outermost shell of an atom of an element is 3d3. The spectral symbol for the ground state is

(a) 4F3/2

(b) 4F9/2

(c) 4D1/2

(d) 4D1/2

Ans. (a)

Sol.

Spectral term = 2S+1LJ = 4F3/2

27. In a spectrum resulting from Raman scattering, let IR denote the intensity of Rayleigh scattering and IS and IAS denote the most intense Stokes line and the most intense anti-Stokes line, respectively. The correct order of these intensities is

(a) IS > R > IAS

(b) IR > IS > IAS

(c) IAS > IR > IS

(d) IR > IAS > IS

Ans. (b)

Sol. Intensity of Rayleigh line is always higher than intensity of stokes and Anti-stokes line. Whereas the intensity of stokes-line is lighter than anti-stokes line

Thus IR > IS > IAS

28. A particle hops randomly from a site to its nearest neighbour in each step on a square lattice of unit lattice constant. The probability of hopping to the positive x -direction is 0.3, to the negative x -direction is 0.2, to the positive y -direction is 0.2 and to the negative y -direction is 0.3 . If a particle starts from the origin, its mean position after N steps is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

29. Let and denote position and momentum operators obeying the commutation relation . If denotes an eigenstate of corresponding to the eigenvalue x, then is

(a) an eigenstate of corresponding to the eigenvalue x

(b) an eigenstate of corresponding to the eigenvalue (x + a)

(c) an eigenstate of corresponding to the eigenvalue (x – a)

(d) not an eigenstate of

Ans. (c)

Sol.

30. The strong nuclear force between a neutron and a proton in a zero orbital angular momentum state is denoted by Fnp(r), where r is the separation between them. Similarly, Fnn(r) and Fpp(r) denote the forces between a pair of neutrons and protons, respectively, in zero orbital momentum state. Which of the following is true on average if the inter-nucleon distance is 0.2 fm < r < 2 fm?

(a) Fnp is attractive for triplet spin state, and Fnn, Fpp are always repulsive

(b) Fnn and Fnp are always attractive and Fpp is repulsive in the triplet spin state

(c) Fpp and Fnp are always attractive and Fnn is always repulsive

(d) All three forces are always attractive

Ans. (b)

Sol. Inside the nucleus the interaction between neutron neutron and newtran-proton is always attractive due to nuclear force whereas between proton-proton it is repulsive due to coulombic interaction:

Thus Fnn and Fnp are always attractive and Fpp is repulsive

31. The Hall coefficient for a semiconductor having both types of carriers is given as

where p and n are the carrier densities of the holes and electrons, µp and µn are their respective mobilities. For a p-type semiconductor in which the mobility of holes is less than that of electrons, which of the following graphs best describes the variation of the Hall coefficient with temperature?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Case I: At low temperature: p >> n, µp < µn

Thus graph (d) correctly repeated the variation of RH with respect to temperature

32. The generator of the infinitesimal canonical transformation and is

(a) q + p

(b) qp

(c)

(d)

Ans. (b)

Sol.

We must check all options but if G = qp

33. Assume that the noise spectral density, at any given frequency, in a current amplifier is independent of frequency. The bandwidth of measurement is changed from 1Hz to 10Hz. The ratio A/ B of the RMS noise current before (A) and after (B) the bandwidth modification is

(a) 1/10

(b)

(c)

(d) 10

Ans. (b)

Sol. Correct option is (b)

34. Let the normalized eigenstates of the Hamiltonian be and . The expectation value and the variance of H in the state are

(a)

(b)

(c)

(d) 2 and 1

Ans. (c)

Sol.

Hence coefficient and in are same so this is not any need to find eigenstate.

35. For a crystal, let denote the energy required to create a pair of vacancy and interstitial defects. If n pairs of such defects are formed, and n << N, N', where N and N' are respectively, the total number of lattice and interstitial sites, then n is approximately

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Thermodynamic probability of such Frienbed defects is

change in entropy is

change in free energy in creating n Frenkel defects

36. In the circuit diagram of a band pass filter shown below, R = 10 .

In order to get a lower cut-off frequency of 150 Hz and an upper cut-off frequency of 10kHz , the appropriate values of C1 and C2 respectively are

(a) 0.1 µF and 1.5 nF

(b) 0.3 µF and 5.0 nF

(c) 1.5 nF and 0.1 µF

(d) 5.0 nF and 0.3 µF

Ans. (a)

Sol.

37. The Bethe-Weizsacker formula for the binding energy (in MeV) of a nucleus of atomic number Z and mass number A is

The ratio Z/A for the most stable isobar of a A = 64 nucleus, is nearest to

(a) 0.30

(b) 0.35

(c) 0.45

(d) 0.50

Ans. (c)

Sol.

given ac = 0.714 and aa = 23.2

38. The phase difference between two small oscillating electric dipoles, separated by a distance d , is . If the wavelength of the radiation is , the condition for constructive interference between the two dipolar radiations at a point P when r >> d (symbols are as shown in the figure and n is an integer) is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Since dipole are in opposite direction, initial phase change will be .

39. The Hamiltonian of two particles, each of mass m, is

, where k > 0 is a constant. The value of the partition function

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

40. In the AC Josephson effect, a supercurrent flows across two superconductors separated by a thin insulating layer and kept at an electric potential difference . The angular frequency of the resultant supercurrent is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

41. A negative muon, which has a mass nearly 200 times that of an electron, replaces an electron in a Li atom. The lowest ionization energy for the muonic Li atom is approximately

(a) the same as that of He

(b) the same as that of normal Li

(c) 200 times larger than that of normal Li

(d) the same as that of normal Be

Ans. (c)

Sol. Ionization energy

For Normal Li-atom

For Muonic Li-atom

42. The wavefunction of a particle of mass m, constrained to move on a circle of unit radius centered at the origin in the xy-plane, is described by , where is the azimuthal angle. All the possible outcomes of measurements of the z-component of the angular momentum Lz in this state, in units of are

(a) ±1 and 0

(b) ±1

(c) ±2

(d) ±2 and 0

Ans. (d)

Sol.

m = 2, –2, 0

43. An alternating current flows through a circular wire loop of radius R, lying in the xy-plane, and centered at the origin. The electric field and the magnetic field are measured at a point such that , where . Which one of the following statements is correct?

(a)

(b)

(c) The time-averaged as a function of the polar angle has a minimum at

(d) is along the azimuthal direction

Ans. (b)

Sol.

44. The positive zero of the polynomial f(x) = x2 – 4 is determined using Newton-Raphson method, using initial guess x = 1. Let the estimate, after two iterations, be x(2). The percentage error is

(a) 7.5%

(b) 5.0%

(c) 1.0%

(d) 2.5%

Ans. (d)

Sol. x0 = 1

45. Which of the following decay processes is allowed?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Thus this is an allowed decay through weak interaction.

46. A metallic wave guide of square cross-section of side L is excited by an electromagnetic wave of wave-number k. The group velocity of the TE11 mode is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

47. A parallel plate capacitor with 1 cm separation between the plates has two layers of dielectric with dielectric constants = 2 and = 4, as shown in the figure below. If a potential difference of 10V is applied between the plates, the magnitude of the bound surface charge density (in units of C/m2) at the junction of the dielectrics is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

48. The Hamiltonian of a system with two degrees of freedom is H = q1p1 – q2p2 + aq12, where a > 0 is a constant. The function q1q2 + p1p2 is a constant of motion only if is

(a) 0

(b) 1

(c) –a

(d) a

Ans. (a)

Sol.

49. The function f(t) is a periodic function of period . In the range , it equals e–t. If denotes its Fourier series expansion, the sum is

(a) 1

(b)

(c)

(d)

Ans. (d)

Sol.

50. The fixed points of the time evolution of a one-variable dynamical system described by are 0.5 and –1. The fixed points 0.5 and –1 are

(a) both stable

(b) both unstable

(c) unstable and stable, respectively

(d) stable and unstable, respectively

Ans. (b)

Sol.

For fixed point yn+1 = yn

51. Following a nuclear explosion, a shock wave propagates radially outwards. Let E be the energy released in the explosion and be the mass density of the ambient air. Ignoring the temperature of the ambient air, using dimensional analysis, the functional dependence of the radius R of the shock front on E, and the time t is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Correct option is (a)

52. The pressure p of a gas depends on the number density of particles and the temperature T as where B2 and B3 are positive constants. Let Tc, c and c denote the critical temperature, critical number density and critical pressure, respectively. The ratio ckBTc/pc is equal to

(a)

(b) 3

(c)

(d) 4

Ans. (b)

Sol.

For critical constants

53. The mean kinetic energy per atom in a sodium vapour lamp is 0.33eV. Given that the mass of sodium is approximately 22.5×109 eV, the ratio of the Doppler width of an optical line to its central frequency is

(a) 7 × 10–7

(b) 6 × 10–6

(c) 5 × 10–5

(d) 4 × 10–4

Ans. (b)

Sol. Dopper shift is

54. In a collector feedback circuit shown in the figure below, the base emitter voltage VBE = 0.7V and current gain for the transistor

The value of the base current IB is

(a) 20 µA

(b) 40 µA

(c) 10 µA

(d) 100 µA

Ans. (a)

Sol. Apply K.V.L in input section

–20V + BIB × 5K + IB × 500K + 0.7V = 0

55. For T much less than the Debye temperature of copper, the temperature dependence of the specific heat at constant volume of copper, is given by (in the following a and b are positive constants)

(a) aT3

(b) aT + bT3

(c) aT2 + bT3

(d)

Ans. (b)

Sol. The specific heat of model is sum of electric and phonon specific heat

C = Ce + Cph