CSIR NET PHYSICS (DEC 2011)
Previous Year Question Paper with Solution.

1. Consider three polarizers P1, P2 and P3 placed along an axis as shown in the Figure.

The pass axis of P1 and P3 are at right angles to each other while the pass axis of P2 makes an angle with that of P1. A beam of unpolarized light of intensity I0 is incident on P1 as shown. The intensity of light emerging from P3 is

(a) 0

(b)

(c)

(d)

Ans. (c)

Sol.

2. A double pendulum consists of two point masses m attached by massless strings of length l as shown in the figure:

The kinetic energy of the pendulum is :

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Equation (2), (3) put in equation (4) then we get

3. In the operational amplifier circuit below, the voltage at point A is

(a) 1.0 V

(b) 0.5 V

(c) 0 V

(d) –5.0 V

Ans. (a)

Sol. Since the current flowing towards node A is zero i.e. I = 0

{VA is potential at point A}

4. A point particle of mass m carrying an electric charge q is attached to a spring of stiffness constant k. A constant electric field E along the direction of the spring is switched on for a time interval T (where T ). Neglecting radiation loss, the amplitude of oscillation after the field is switched off is:

(a) qE/k

(b) qET2/m

(c)

(d)

Ans. (d)

Sol. Equation of Motion:

5. A constant force F is applied to a relativistic particle of rest mass m. If the particle starts from rest at t = 0, its speed after a time t is

(a) Ft/m

(b)

(c) c(I – e–Ft/mc)

(d)

Ans. (d)

Sol.

Equation (1), (2) put in equation (3)

6. The potential of a diatomic molecule as a function of the distance r between the atoms is given by . The value of the potential at equilibrium separation between the atoms is:

(a) –4a2/b

(b) –2a2/b

(c) –a2/2b

(d) –a2/4b

Ans. (d)

Sol.

7. Four equal point charges are kept fixed at the four vertices of a square. How many neutral points (i.e., points where the electric field vanishes will be found inside the square?

(a) 3

(b) 4

(c) 5

(d) 7

Ans. (c)

Sol. One neutral point is at the centre of the square. By symmetry of the square, there are four more points, so in total there are 5 neutral points.

8. A static charge distribution gives rise to an electric field of the form , where and R are positive constants. The charge contained within a sphere of radius R, centred at the origin is :

(a)

(b)

(c)

(d)

Ans.

Sol.

We know that (poission equation)

No correct option is given. All options have incorrect dimension. has dimension of charge, which is correct according to our calculation.

9. A counter consists of four flip-flops connected as shown in the figure.

If the counter is initialized as A0 A1 A2 A3 = 0110, the state after the next clock pulse is

  1. 1000

(b) 0001

(c) 0011

(d) 1100

Ans. (b)

Sol.

After one cycle

10. In a Young's double slit interference experiment, the slits are at a distance 2L from each other and the screen is at a distance D from the slits. If a glass slab of refractive index µ and thickness d is placed in the path of one of the beams, the minimum value of d for the central fringe to be dark is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Extra path difference = (µ – 1)d; here, path difference needed to shift maxima to minima

11. The pins 0, 1, 2, and 3 of part A of a microcontroller are connected with resistors to drive an LED at various intensities as shown in the figure.

For Vcc = 4.2V and a voltage drop of 1.2 V across the LED, the range (maximum current) and resolution (Step size) of the drive current are, respectively,

(a) 4.0 mA and 1.0 mA

(b) 15.0 mA and 1.0 mA

(c) 7.5 mA and 0.5 mA

(d) 4.0 mA and 0.5 mA

Ans. (c)

Sol. Potential at point P will be 4.2 – 1.2 = 3.0 V. For maximum current value of resistance should be minimum.

So, all 4 resistors will be in parallel

For minimum current maximum resistance will be taken,

12. An unbiased dice is thrown three times successively. The probability that the numbers of dots on the uppermost surface add up to 16 is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The combination that add upto 16 are (6, 5, 5), (5, 5, 6), (5, 6, 5), (6, 6, 4), (6, 4, 6), (5, 6, 6)

Number of successful events = 6; Number of events in sample space = 63 = 216

13. The generating function for the Legendre polynomials Pn(x) is F(x, t) = (1 – 2xt + t2)–1/2. The value of P3(–1) is:

(a) 5/2

(b) 3/2

(c) +1

(d) –1

Ans. (d)

Sol.

Putting x = –1, we get

Comparing coefficient of t3 on both side, P3 (–1) = –1.

14. Given that the ground state energy of the hydrogen atom is 13.6 eV, the ground state energy of positronium (which is a bound state of an electron and a position) is

(a) + 6.8 eV

(b) –6.8 eV

(c) –13.6 eV

(d) –27.2 eV

Ans. (b)

Sol. For H atom,

where RM = Rydberg constant which is proportional to reduced mass µ of the system.

For positronium,

Energy of positronium atom,

Ground state energy = –6.8 eV

15. Two particles of identical mass move in circular orbits under a central potential . Let and be the angular momenta and r1, r2 be the radii of the orbits respectively. If , the value of r1/r2 is:

(a)

(b)

(c) 2

(d) 1/2

Ans. (a)

Sol.

16. The equation of the plane that is tangent to the surface xyz = 8 at the point (1, 2, 4) is

(a) x + 2y + 4z = 12

(b) 4x + 2y + z = 12

(c) x + 4y + 2z = 12

(d) x + y = z = 7

Ans. (b)

Sol. Consider a point Q(x, y, z) on the tangent plane to the surface xyz = 8 at point P(1, 2, 4).

Therefore, lies on tangent plane.

17. The energy of the first excited quantum state of a particle in the two-dimensional potential is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

For ground state

For first excited state, nx = 1, ny = 0

18. The internal energy E of a system is given by , where b is a constant and other symbols have their usual meaning. The temperature of this system is equal to

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

19. Consider a praticle in a one dimensional potential that satisfies V(x) = V(–x). Let and denote the ground and the first excited states, respectively, and let be a normalized state with and being real constants. The expectation value of the position operator x in the state is given by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. For a symmetric potential, has definite party.

as 'x' is odd operator and has definite parity. (Integral of even × odd function is zero) and similarly .

20. A 3 × 3 matrix M has Tr[M] = 6, Tr[M2] = 26 and Tr[M3] = 90. Which of the following can be a possible set of eigenvalues of M ?

(a) {1, 1, 4}

(b) {–1, 0, 7}

(c) {–1, 3, 4}

(d) {2, 2, 2}

Ans. (c)

Sol. Since, sum of eigenvalues of the matrix = Trace of the matrix.

It is trivial to check that only {–1, 3, 4} satisfies these three equation.

21. The perturbation H' = bx4, where b is a constant, is added to the one dimensional harmonic oscillator potential . Which of the following denotes the correction to the ground state energy to first order in b ?

[Hint : The normalized ground state wave function of the one dimensional harmonic oscillator potential is . You may use the following integral

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Comparing with the given standard integral and n = 2

22. A battery powers two circuits C1 and C2 as shown in the figure

The total current I drawn from the battery is estimated by measuring the currents I1 and I2 through the individual circuits. If I1 and I2 are both 200 mA and if the errors in their measurement are 3 mA and 4 mA respectively, the error in the estimate of I is:

(a) 7.0 mA

(b) 7.5 mA

(c) 5.0 mA

(d) 10.5 mA

Ans. (c)

Sol. P = P1 + P2

Error in estimate of I.

23. Consider a Maxwellian distribution of the velocity of the molecules of an ideal gas. Let Vmp and Vrms denote the most probable velocity and the root mean square velocity, respectively. The magnitude of the ratio Vmp/Vrms is:

(a) 1

(b) 2/3

(c)

(d) 3/2

Ans. (c)

Sol.

24. If the number density of a free electron gas in three dimensions is increased eight times, its Fermi temperature will

(a) increase by a factor of 4

(b) decrease by a factor of 4

(c) increase by a factor of 8

(d) decrease by a factor of 8

Ans. (a)

Sol.

25. A system of N non-interacting spin-½ particles is placed in an external magnetic field H. The behaviour of the entropy of the system as a function of energy is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. There are N non-interacting particles.

Magnetic moment of each particle = µ

Energy of particle when aligned parallel with magnetic field

Energy of particle when aligned antiparallel with magnetic field

Where

Total energy E of the isolated system is

Number of Microstates,

= (N ln N – N) – (N ln N – N) – (N+ ln N+ – N+)

(We have used Stirling approximation)

= – [N ln (N/N) + N+ (N+ / N)]

where E = 0, S = N k ln 2

The equation of S versus E represents the following curve.

26. A particle of mass 'm' moves inside a bowl. If the surface of the bowl is given by the equation , where a is a constant, the Lagrangian of the particle is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Eq. (1), (4) put in Eq. (5)

27. An electron of energy 27 GeV collides with a proton of energy 820 GeV. The heaviest particle which can be produced in this collision has mass close to

(a) 300 GeV

(b) 821 GeV

(c) 850 GeV

(d) 1127 GeV

Ans. (a)

Sol. Electron-proton interaction is mediated by Z-particles which has mass of 91 GeV. The total energy available is 820 + 27 = 847 GeV, from where the Z-particle carries 91 GeV. The remaining 756 GeV in available to produce a jet of hadrons. Only option (a) is correct because 300 GeV < 756 GeV.

28. Let x1(t) and x2(t) be two linearly independent solutions of the differential equation , and let . If w(0) = 1, then w(1) is given by

(a) 1

(b) e2

(c) 1/e

(d) 1/e2

Ans. (d)

Sol. Given, differential equation is of the form :

29. Assume that the free energy of a magnetic system has an expansion in the order parameter M of the form F(M, T) = a(T – TC)M2 + bM4 + cM6, with a, b and c > 0. As the temperature is lowered below TC, the system undergoes a phase transition. The behaviour of the order parameter just below the transition, where (T – TC) is very small, is best described by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Near the transition point, we can consider only the first two term F = a (T – Tc)M2 + bM4

For this equation to be satisfied,

30. A planet of mass m moves in the inverse square central force field of the Sun of mass M. If the semi-major and semi-minor axes of the orbit are a and b, respectively, the total energy of the planet is :

(a)

(b)

(c)

(d)

Ans.

Sol. None of the option is correct. Correct answer is .

31. Let and denote the normalized eigenstates corresponding to the ground and first excited state of a one dimensional harmonic oscillator. The uncertainty in the state , is:

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

32. A laser operating at 500 nm is used to excite a molecule. If the Stokes line is observed at 770 cm–1, the approximate positions of the Stokes and the anti-Stokes lines are

(a) 481.5 nm and 520 nm

(b) 481.5 nm and 500 nm

(c) 500 nm and 520 nm

(d) 500 nm and 600 nm

Ans. (a)

Sol. Stokes and anti-S. According to the shell tokes lines never occur at the Laser frequency, which is 500 nm. Only option (a) does not have this number.

33. The graph of the function

(Where n = 0, 1, 2, ......) is shown below.

Its Laplace transform is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Given function f(x) is a periodic function of period T = 2

34. The energy levels of electrons of mass 'm' and charge 'e' confined in an area A in the xy-plane with a uniform magnetic field B applied in the z-direction are given by , n = 0, 1, 2, ....... The degeneracy of each level is . The lowest level is completely filled the other are empty. The fermi energy , where N is the total number of electrons, is:

(a) coincident with the n = 0 level

(b) coincident with the n =1 level

(c) midway between the n = 0 and the n = 1 levels

(d) midway between the n = 1 and the n = 2 levels.

Ans. (d)

Sol.

{But in the question Fermi energy is given as which is incorrect. Therefore, this question is incorrect}.

Number of zero field states in an interval is

The level n = 1.5 is between 1 and 2.

35. An annulus of mass M made of a material of uniform density has inner and outer radii 'a' and 'b' respectively. Its principal moment of inertia along the axis of symmetry perpendicular to the plane of the annulus is:

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

We know that

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

36. The principal value of the real integral is:

(a)

(b)

(c)

(d) 0

Ans. (d)

Sol. , the integrand blows up at x = –1, and at x = –2 but its principal value can be calculated by using contour integration, has poles are on the real axis and lie at z = –1 and z = –2.

Res f (z = –2) = –1

37. The minimum energy of an electron (the rest mass of which is 0.5 MeV) that can emit Cherenkov radiation while passing through water (of refractive index 1.5) is approximately

(a) 1.0 MeV

(b) 3.0 MeV

(c) 0.6 MeV

(d) 0.5 MeV

Ans. (c)

Sol.

E = 0.67 MeV which is close to the approximate answer given.

38. A heater and a thermocouple are used to measure and control temperature T of a sample at T0 = 250ºC. A feedback circuit supplies power 'P' to the heater according to the equation with appropriately tuned values of the coefficients G and D. In order to maintain temperature stability in the presence of an external heat perturbation which causes small but rapid fluctuations of temperature, it is necessary to

(a) decrease D

(b) increase D

(c) decrease G

(d) increase G

Ans. (a)

Sol.

If increases, D must decrease.

39. The trajectory on the zpz-plane (phase-space trajectory) of a ball bouncing perfectly elastically off a hard surface at z = 0 is given by approximately by (neglect friction) :

(a)

(b)

(c)

(d)

Ans. (a)

Sol. When object is moving upward direction then momentum of the object is decreasing.

At z = z0, P = 0 (minima)

When object is moving downward direction then momentum of the object is increasing.

At z = 0, Pz = maxima

40. According to the shell model, the spin and parity of the two nuclei and are, respectively.

(a)

(b)

(c)

(d)

Ans. (d)

Sol. For odd proton

Shell model energy levels, 1s1/2 1p3/2 1p1/2 1d5/2 2s1/2 1d3/2 1f7/2 2p3/2 1f5/2 2p1/2 1g9/2 1g7/2

Therefore, have odd proton, last proton will to to 1g7/2 (l = 4)

So, Spin-parity

For odd Neutron, last neutron will go to 1g7/2 (l = 4)

So, spin-parity . So then are is

41. The wave function of a particle at time t = 0 is given by , where and are the normalized eigenstates with eigenvalues E1 and E2 respectively, (E2 > E1). The shortest time after which will become orthogonal to is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

42. A gas of N non-interacting particles is in thermal equilibrium at temperature T. Each particle can be in any of the possible non-degenerate states of energy 0, and 4. The average energy per particle of the gas, when , is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Since, we retain only the first term of exponentials.

43. The first few terms in the Taylor series expansion of the function f(x) = sin x around are

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

f(x) = sin x, f '(x) = cos x, f "(x) = – sin x, f "'(x) = – cos x

44. A one-dimensional chain consists of a set of N rods each of length a. When stretched by a load, each rod can align either parallel or perpendicular to the length of the chain. The energy of a rod is – when aligned parallel to the length of the chain and is + when perpendicular to it. When the chain is in thermal equilibrium at temperature T, its average length is:

(a) Na/2

(b) Na

(c)

(d)

Ans. (c)

Sol. The mean length of the segment is

Average length

45. If the hyperfine interaction in an atom is given by Where and denote the electron and proton spins, respectively, the splitting between the 3S1 and 1S0 state is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

Splitting, (For singlet s = 0, For triplet s = 1)

46. Consider a solenoid of radius R with n turns per unit length, in which a time dependent current (where R/c << 1) flows. The magnitude of the electric field at a perpendicular distance r < R from the axis of symmetry of the solenoid, is:

(a) 0

(b)

(c)

(d)

Ans. (d)

Sol. We know that

From (1) and (2)

From solenoid

Equation (4) put in Equation (3) than we get

47. The difference in the Coulomb energy between the mirror nuclei and is 6.0 MeV. Assuming that the nuclei have a spherically symmetric charge distribution and that e2 is approximately 1.0 MeV-fm, the radius of the nucleus is

(a) 4.9 × 10–13 m

(b) 4.9 × 10–15 m

(c) 5.1 × 10–13 m

(d) 5.1 × 10–15 m

Ans. (b)

Sol. Coulomb energy of uniformly charged sphere of radius,

48. The ratio of intensities of the D1 and D2 lines of sodium at high temperature

(a) 1 : 1

(b) 2 : 3

(c) 1 : 3

(d) 1 : 2

Ans. (d)

Sol. D1 line : line :

The intensites will depend on the population of the levels

At high temperature the intensify ratio is determined by the ratio of degeneracies of 2P3/2 and 2P1/2

49. A constant perturbation as shown in the figure below acts on a particle of mass m confined in a infinite potential well between 0 and L.

The first-order correction to the ground state energy of he particle is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

50. An atom of mass M can be excited to a state of mass M + by photon capture. The frequency of a photon which can cause this transition is :

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Neglecting P << Mc we get,

51. The excitation of a three-dimensional solid are bosonic in nature with their frequency and wave-number k are related by in the large wavelength limit. If the chemical potential is zero, the behaviour of the specific heat of the system at low temperature is proportional to

(a) T1/2

(b) T

(c) T3/2

(d) T3

Ans. (c)

Sol.

52. Gas molecules of mass 'm' are confined in a cylinder of radius R and height L (with R >> L) kept vertically in the Earth's gravitational field. The average energy of the gas at low temperatures (such that mgL >> kBT) is given by

(a) NkBT/2

(b) 3NkBT/2

(c) 2NkB/T

(d) 5NkBT/2

Ans. (d)

Sol. Average energy = (Number of degrees of freedom)

For translational motion, number of degrees of freedom = 3.

, V = mgz (Only one degree of freedom)

According to Virial theorem,

For N-particles,

53. The figure below shows a voltage regulator utilizing a Zener diode of breakdown voltage 5V and a positive triangular wave input of amplitude 10V.

For Vi > 5V, the Zener regulates the output voltage by channeling the excess current through it self. Which of the following waveforms shows the current 'i' passing through the Zener diode?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. [As Zener diode is not ideal]

When Vi < 5 [upto t = 1 sec, 3 < t < 5]

Then Zener is reverse biased hence, very-2 small current will pass through zener diode.

Then i = very small fixed current will flow.

Now, when Vi < 5.

Then Zener comes in breakdown then maximum current will flows through it.

1 will be short circuited at 1 sec.

which will rise steadily as it is not ideal.

54. A constant electric current I in an infinitely long straight wire is suddenly switched on at t = 0. The vector potential at a perpendicular distance r from the wire is given by . The electric field at a distance r(< ct) is:

(a) 0

(b)

(c)

(d)

Ans. (d)

Sol.

55. Monochromatic light of wavelength 660 nm and intensity 100 mW/cm2 falls on a solar cell of area 30 cm2. The conversion efficiency of the solar cell is 10%. If each converted photon results in an electron-hole pair, what is the maximum circuit current supplied by the solar cell? (Take h = 6.6 × 10–34J-s, c = 3 × 108 m/s and e = 1.6 × 10–19 C).

(a) 160 mA

(b) 320 mA

(c) 1600 mA

(d) 3200 mA

Ans. (a)

Sol. Intensity fallen = 100 mw × 30 = 3 watt in the cell.

If the efficiency is 10%. Then, 0.3 watt power is carried out by the current.

The maximum potential generated

Therefore, the maximum current