PYQ 2003 - VedPrep

GATE PHYSICS 2003
Previous Year Question Paper with Solution.

1. The two vectors are

(a) related by a rotation

(b) related by a reflection through the xy-plane

(d) not linearly independent

Ans. (a)

Sol.

2. A 3 × 3 matrix has eigenvalues 0, 2 + i and 2 – i. Which one of the following statements is correct?

(a) The matrix is Hermitian

(b) The matrix is unitary

(d) The determinant of the matrix is zero

Ans. (d)

Sol. Determinant of matrix = product of the eigenvalues = 0

3. The value of the integral , where z is a complex variable and C is the unit circle with the origin as its centre, is

(a) 0

(b)

(c)

(d)

Ans. (a)

Sol.

4. A particle with an initial velocity v₀i enters a region with an electric field E₀j and a magnetic field B₀j. The trajectory of the particle will

(a) be an ellipse

(b) be a cycloid

(d) not be confined to any plane

Ans. (d)

Sol. Total force acting on particle

So, particle will not be confined to any plane.

5. An object of mass m rests on a surface with coefficient of static friction µ. Which of the following is NOT correct?

(a) The force of friction is exactly µmg

(b) The maximum force of friction is µmg

(d) The force of friction opposes any effort to move the object

Ans. (a)

Sol. Static friction is given as f_s< µN

6. The Lagrangian of a particle of mass m moving in a plane is given by L = (1/2) where v_x and v_y are velocity components and a is a constant. The canonical momenta of the particle are given by

(a) p_x = mv_x and p_y = mv_y

(b) p_x = mv_x + ay and p_y = mv_y + ax

(d) p_x = mv_x – ay and p_y = mv_y – ax

Ans. (c)

Sol.

7. Two events are separated by a distance of 6 × 10⁵ km and the first event occurs 1's before the second event. The interval between the two events

(a) is time-like

(b) is light-like (null)

(d) cannot be determined from the information given

Ans. (c)

Sol. The distance between two event is

The time difference between two event is

Therefore, interval between the events is space like.

8. An electric charge, +Q is placed on the surface of a solid, conduction sphere of radius a. The distance measured from the centre of the sphere is denoted as r. Then

(a) the charge gets distributed uniformly through the volume of the sphere

(b) the electrostatic potential has the same value for r < a

(d) the electric field is given by for r < a

Ans. (b)

Sol. As soon as we put same charge on the sphere it will distribute uniformly over the whole surface. So, field inside the sphere will be zero. As a result potential will be constant.

9. An electric field applied along the length of a long cylinder produces a polarization P. The depolarization field produced in this configuration is

(a)

(b)

(c)

(d) 0

Ans. (d)

Sol. Let be the surface charge density on the cross-sectional area that appears due to polarization. So, the depolarization field be

10. Which one of the following Maxwell's equations implies the absence of magnetic monopoles?

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

So, magnetic flux through a close surface is zero.

This implies that magnetic monopoles do not exist.

11. An electromagnetic wave is propagating in free space in the z-direction. If the electric field is given by where then the magnetic field is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. In free space, for progressive wave, we have

12. Given a wave with the dispersion relation for k > 0 and m > 0, which one of the following is true?

(a) The group velocity is greater than the phase velocity

(b) The group velocity is less than the phase velocity

(d) There is no definite relation between the group velocity and the phase velocity

Ans. (b)

Sol.

13. Which of the following is a valid normalized wave function for a particle in a one dimensional infinite potential well of width L centered at x = 0?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Normalized wave function for a particle in a one dimensional infinite potential well of width L centered at x = 0, is

14. The commutator [x, P²], where x and P are position and momentum operators respectively, is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

15. A spin half particle is in the state . The expectation values of are given by

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

16. The spectral term for the atom with 70% filled subshell and only S = 3/2 is

(a) ³P₀

(b) ⁴F_9/2

(d) ⁴P_1/2

Ans. (b)

Sol. For , means either 3 electrons of 3 unpaired subshells. As the shell is 70% filled, therefore, it can not have only three electrons. Thus we have seven d-electrons. Thus

According to the Hund's rule for more than half filled shell, we take highest value of J for ground state.

17. The hyperfine splitting of the spectral lines of an atom is due to

(a) the coupling between the spins of two or more electrons

(b) the coupling between the spins and the orbital angular momenta of the electrons

(d) the effect of external electromagnetic fields

Ans. (c)

Sol. The hyperfine splitting of the spectral lines of an atom is due to the coupling between the electronspin and the nuclear spin.

18. A piston containing an ideal gas is originally in the state X (see figure). The gas is taken through a thermal cycle XYX as shown. The work done by the gas is positive if the direction of the thermal cycle is

(a) clockwise

(b) counter-clockwise

(d) clockwise from XY and counter-clockwise from YX

Ans. (a)

Sol. Work done = Area between P-V curve and volume axis.

So, W_XY = Area XUABX and W_YX = Area YXBAY

Total work done, W = W_XY + W_YX

Since, W_XY is positive as volume is increased and W_YX is negative as volume is decreased.

So, W = Area XYABX – Area YXBAY

Since Area XYABX > Area YXBAY, W is positive.

So work done by the gas is positive if the direction of the thermal cycle is clockwise.

19. A second order phase transition is one in which

(a) the plot of entropy as a function of temperature shows a discontinuity

(b) the plot of specific heat as a function of temperature shows a discontinuity

(d) the plot of comprehensibility as a function of temperature is continuous

Ans. (b)

Sol. The second derivative of Gibbs free energy with respect to temperature and pressure are discontinuous in second order phase transition.

i.e.

So, isothermal compressiblility and specific heat at constant volume (Cv) are discontinuous in second order phase transition.

20. Consider the Fermi-Dirac distribution function f(E) at room temperature (300 K) where E refers to energy. If E_F is the Fermi energy, which of the following is true?

(a) f(E) is a step function

(b) f(E_F) has a value of 1/2

(d) f(E) is large and tends to infinity as E decreases much below E_F

Ans. (b)

Sol.

21. If the ionic radii of Mn and S are 0.80 and 0.184 nm respectively, the structure of MnS will be

(a) cubic closed packed

(b) body centered cubic

(d) primitive cubic cell

Ans. (c)

Sol. Ionic solid MnS will have NaCl type crystal structure.

22. A cubic cell consists of two atoms of masses m₁ and m₂ (m₁ > m₂) with m₁ and m₂ atoms situated on alternate planes. Assuming only nearest neighbor interactions, the centre of mass of the two atoms

(a) moves with the atoms in the optical mode and remains fixed in the acoustic mode

(b) remains fixed in the optical mode and moves with the atoms in the acoustic mode

(d) moves with the atoms in both optical and acoustic modes

Ans. (b)

Sol. For optical brands: . This indicates that the two atoms move in opposite directions and their amplitudes are inversely proportional to their masses so that the centre of mass of the unit cell remains unchanged.

For acoustical branch: . This means that the two atoms of different masses move in the same direction with the same amplitude and there is a movement of their centers of masses as well.

23. In simple metals the phonon contribution to the electrical resistivity at temperature T is

(a) directly proportional to T above Debye temperature and to T³ below it

(b) inversely proportional to T for all temperatures

(d) directly proportional to T above Debye temperature and to T⁵ below it

Ans. (d)

Sol. The phonon contribution to the electrical resistivity at temperature T varies as:

(at temperatures higher than Debye temperature i.e. T > ).

and (at temperatures lower than Debye temperature i.e. T < ).

24. The effective mass of an electron in a semiconductor can be

(a) negative near the bottom of the band

(b) a scalar quantity with a small magnitude

(d) negative near the top of the band

Ans. (a)

Sol. The effective mass (m*) can be represented as

From this figure, it is clear that m* is negative near the top of the band.

25. The dielectric constant of water is 80. However its refractive index is 1.75 invalidating the expression . This is because

(a) the water molecule has a permanent dipole moment

(b) the boiling point of water is 100ºC

(d) water is transparent to visible light

Ans. (a)

Sol. Water is a polar molecule or dielectric material. Its molecule has a permanent dipole moment that's why its dielectric constant and refractive index do not validate the expression .

26. The nucleus of the atom ⁹Bc₄ consists of

(a) 13 up quarks and 13 down quarks

(b) 13 up quarks and 14 down quarks

(d) 14 up quarks and 14 down quarks

Ans. (b)

Sol. 4Be⁹ number of protons (Z) = 4 and number of neutrons (N) = 9 – 4 = 5

For one proton, there will be uud (quark structure) and for one neutron, there will be udd (quark structure).

For 4 proton, (4u, 4u, 4d) and for 5 neutrons (5u, 5d, 5d).

Therefore, total up quarks for ⁴Be⁹ = 13 and total down quarks for ₄Be⁹ = 14

27. Which one of the following nuclear reactions is possible?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The basic equations of -disintegrations is

¹⁷N₇ has 7 protons and 6 neutrons, ¹³C₆ has 6 protons and 7 neutrons. So, one proton is changed into neutron.

28. Suppose that a neutron at rest in free space decays into a proton and an electron. This process would violate

(a) conservation of charge

(b) conservation of energy

(d) conservation of angular momentum

Ans. (d)

Sol. In the process of neutron decay into proton and electron, the conservation of angular momentum will be violated (according to Fermi theory)

29. Which one of the following is true for a semiconductor p^–n junction with no external bias?

(a) The total charge in the junction is not conserved

(b) The p side of the junction is positively charged

(d) No charge develops anywhere in the junction

Ans. (c)

Sol. At the pn-junction of a semiconductor, when no external bias, due to diffusion there will be no majority charge carries, but negative minority carrier on the p-side and positive minority carrier on the n-side.

30. Which one of the set of values given below does NOT satisfy the Boolean relation R = PQ' (where Q' denotes NOT Q)?

(a) P = 1, Q = 1, R = 0

(b) P = 1, Q = 1, R = 1

(d) P = 0, Q = 1, R = 1

Ans. (b, d)

Sol. By putting the value of given option check which satisfies the relation

So, (b) and (d) is not satisfying.

31. The curl of the vector A = zi + xj + yk is given by

(a) i + j + k

(b) i – j + k

(d) – i – j – k

Ans. (a)

Sol.

32. Consider the differential equation d²x/dt² + 2dx/dt + x = 0. At time t = 0, it is given that x = 1 and dx/dt = 0. At t = 1, the value of x is given by

(a) 1/e

(b) 2/e

(d) 3/e

Ans. (b)

Sol.

Let, x = C.e^mt be the trial solution.

Therefore, solution will be x = (C₁ + C₂t)e^–t

33. S_ij and A_ij represent a symmetric and an antisymmetric real-valued tensor respectively in three dimensions. The number of independent components of S_ij and A_ij are

(a) 3 and 6 respectively

(b) 6 and 3 respectively

(d) 9 and 6 respectively

Ans. (b)

Sol. Number of independent components of a symmetric tensor of rank 2 in 3-D will be .

Number of independent components of an anti-symmetric tensor of rank w in 3-D will be .

34. Consider the four statements given below about the function f(x) = x⁴ – x² in the range . Which one of the following statements is correct?

P the plot of f(x) versus x has two maxima and two minima

Q the plot of f(x) versus x cuts the x axis at four points

R the plot of f(x) versus x has three extrema

S no part of the plot f(x) versus x lies in the fourth quadrant

Pick the right combination of correct choices from those given below

(a) P and R

(b) R only

(d) P and Q

Ans. (b)

Sol. f(x) = x⁴ – x²

f '(x) = 4x² – 2x

For maximum or minimum of f(x) : f '(x) = 0

Now, f "(x) = 12x² – 2

f "(x = 0) = –2 < 0. So, at x = 0 f(x) has a maximum value

f(x) has a minimum value

So, f(x) has three extremum points i.e. statement R is correct.

When plot of f(x) vs x, cuts x-axis, then f(x) = 0

For 0 < x < 1, x⁴ < x² and f(x) < 0

So, the corresponding part of the plot lies in the 4^th quadrant.

35. The Fourier transform of the function f(x) is . The Fourier transform of df(x)/dx is

(a) dF(k)/dk

(b)

(d) ikF(k)

Ans. (c)

Sol.

36. A particle of mass m is moving in a potential of the form (3x² + 3y² + 2z² + 2xy). The oscillation frequencies of the three normal modes of the particle are given by

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

37. The speed of a particle whose kinetic energy is equal to its rest mass energy is given by (c is the speed of light in vacuum)

(a) c/3

(b)

(d)

Ans. (d)

Sol. Kinetic energy = m₀c²

38. Electromagnetic waves are propagating along a hollow, metallic waveguide whose cross-section is a square of side W. The minimum frequency of the electromagnetic waves is

(a) c/W

(b) 2c/W

(c)

(d)

Ans. (c)

Sol. The cut-off frequency of electromagnetic wave inside a waveguide is given by

For square cross-section, a = b = W

Therefore, the minimum is (take m = 1, n = 0)

39. Consider the given statements about E(r, t) and B(r, t), the electric and magnetic vectors respectively in a region of free space

P Both E and B are conservative vector fields

Q Both E and B are central force fields

R E and B are mutually perpendicular in the region

S Work done by B on a moving charge in the region is zero

Choose the right combination of correct statements from the following

(a) P and R

(b) R and S

(d) P and Q

Ans. (c)

Sol. Since, the displacement of the charge particle is always perpendicular to the magnetic force. So, work done by the magnetic field is zero.

40. An infinite conducting sheet in the x-y plane carries a surface current density K along the y-axis. The magnetic field B for z > 0 is

(a) B = 0

(b) B = µ₀Kk/z

(d) B = µ₀Kj/(x² + z²)^0.5

Ans. (c)

Sol.

41. A parallel beam of infrared radiation of wavelength of 1.01 × 10^–6 m is incident normally on a screen with two slits 5 × 10^–6 m apart and the resulting interference pattern is observed on a distant screen. What is the largest number of maxima that can be observed on the screen?

(a) 4

(b) 9

(d) infinitely many

Ans. (a)

Sol. For nth order maxima

42. A parallel beam of electrons of a given momentum pass through a screen S₁ containing a slit and then produces a diffraction pattern on a screen S₂ placed behind it. The width of the central maximum observed on the screen S₂ can be increased by

(a) decreasing the distance between the screens S₁ and S₂

(b) increasing the width of the slit in screen S₁

(d) increasing the momentum of the electrons

Ans. (c)

Sol. Condition for the first order minima is , where a is the slit width and is the angle of diffraction for first order minima. Width of the central maximum is given by,

where D is the distance between the slit and screen and p is momentum of the electrons.

Thus the width of the central maxima can be increased by one of the following ways: (i) by increasing D, (ii) by decreasing p, (iii) decreasing a.

43. An electron in a time independent potential is in a state which is the superposition of the ground state (E₀ = 11eV) and the first excited state (E₁ = 1eV). The wave function of the electron will repeat itself with a period of

(a) 3.1 × 10^–18 s

(b) 2.1 × 10^–15 s

(d) 1.0 × 10^–9 s

Ans. (b)

Sol. Given wave function of the particle:

Wave function of the particle at time 't' will be

Wave function of the particle will repeat it self after time

44. A particle has the wave function . Which one of the following is correct?

(a) This is an eigenstate of both energy and momentum

(b) This is an eigenstate of momentum and not energy

(d) This is not an eigenstate of energy of momentum

Ans. (b)

Sol.

45. A free particle with energy E whose wave-function is a plane wave with wavelength enters a region of constant potential V > 0 where the wavelength of the particle is . The ratio (V/E) is

(a) 1/2

(b) 2/3

(d) 4/5

Ans. (c)

Sol. Wavelength of the particle in Region-I:

Wavelength of the particle in Region-II:

46. The vibrational spectrum of a molecule exhibits a strong line with P and R branches at a frequency v₁ and a weaker line at a frequency v₂. The frequency v₃ is not shown up. Its vibrational Raman spectrum shows a strongly polarized line at frequency v₃ and no feature at v₁ and v₂.

(a) the molecule could be linear

(b) the molecule lacks a center of inversion

(d) v₃ arises from a bending mode

Ans. (a)

Sol. Since, the frequencies do not coincide in Raman and vibrational spectra thus by mutual exclusion principle molecule should have centre of symmetry. This could be satisfied by linear molecule e.g. CO₂(O=C=O).

47. Three values of rotational energies of molecules are given below in different units

P 10 cm^–1

Q 10^–23 J

R 10⁴ MHz

Choose the correct arrangement in the increasing order of energy

(a) P, Q, R

(b) R, Q, P

(d) Q, R, P

Ans. (b)

Sol.

For Q: E = 10^–23 J

For R : Given frequency v = 10⁴ MHz = 10⁴ × 10⁶ Hz

Energy E = hv = 6.63 × 10^–34 J.S × 10⁴ × 10⁶ s^–1 = 6.63 × 10^–24 J

Thus, increasing order of energy is

19.89 × 10^–23 J > 10^–23 J > 6.63 × 10^–24 J

48. The short wavelength cut off of the continuous X-ray spectrum from a nickel target is 0.0825 nm. The voltage required to be applied to an X-ray tube is

(a) 0.15 KV

(b) 1.5 KV

(d) 150 KV

Ans. (c)

Sol.

49. The spin-orbit coupling constant for the upper state of sodium atom which emits D lines of wave numbers 16956.2 and 16973.4 cm^–1 is

(a) 15 cm^–1

(b) 11.4 cm^–1

(d) 15.1 cm^–1

Ans. (b)

Sol. The spin-orbit interaction energy is given by

(for one electron atom)

where a is coupling constant.

Ground state configuration of sodium spectrum (that emit D line) is ²S. So

Ground state energy level is ²S_1/2. Since there is only one J value. Therefore, there will be no splitting for ²S_1/2.

Excited state term is ²P, i.e.

This level will split into two energy level ²P_1/2, ²P_3/2.

For ²P_1/2 : Let a' be the split-coupling constant for ²P level.

Therefore energy level is

For the first lines transition

D2 = 16973.4 cm^–1

For the second lines transition

From equation (1) and (2),

50. Consider the following statements about molecular spectra

P CH₄ does not give pure rotational Raman lines

Q SF₆ could be studied by rotational Raman spectroscopy

R N₂ shows infrared absorption spectrum

S CH₃CH₃ shows vibrational Raman and infrared absorption lines

T H₂O₂ shows pure rotational spectrum

Choose the right combination of correct statements

(a) P and Q

(b) P, R and T

(d) Q and R

Ans. (c)

Sol. (P) for pure rotational Raman spectrum molecule should have permanent electric dipolemoment but CH₄ does not have permanent dipole. So, it will not show pure rotational Raman spectrum.

(Q) SF₆ also has not permanent dipole moment. So, it will not show pure rotational spectrum.

(R) IR-absorption occurs from the stretching and bending at the covalent bonds in molecules. Since, N₂ has symmetric bonds and bond streching does not change the dipole moment of the molecule. So, it will not show IR spectrum.

(S) For vibrational Raman and infrared absorption lines are observed if the polarisability of the molecule change as molecule vibrates. C₂H₆ follow the above condition.

(T) Since, H₂O₂ has permanent dipole moment so it will shows the pure rotational spectrum

51. The temperature of a cavity of fixed volume is doubled. Which of the following is true for the black-body radiation inside the cavity?

(a) its energy and the number of photons both increase 8 times

(b) its energy increases 8 times and the number of photons increases 16 times

(d) its energy and the number of photons both increase 16 times

Ans. (c)

Sol. In case of photons, and density of states is

So the average energy in cavity,

Also, the number of photons are given by

From equations (i) and (ii), if temperature is doubled, then and

52. A sample of ideal gas with initial pressure P and volume V is taken through an isothermal expansion proceed during which the change in entropy is found to be . The universal gas constant is R. Then the work done by the gas is given by

(a)

(b)

(d)

Ans. (a)

Sol. For isothermal process, dT = 0

dU = 0 for an ideal gas

From first law of thermodynamics,

53. Hydrogen molecules (mass m) are in thermal equilibrium at a temperature T. Assuming classical distribution of velocity, the most probable speed at room temperature is

(a) (k_BT)/m

(b) 2(k_BT)/m

(c)

(d)

Ans. (c)

Sol. The Maxwell-Boltzmann distribution is given by

54. Consider the energy E in the first Brillouin zone as a function of the magnitude of the wave vector k for a crystal of lattice constant a. Then

(a) the slope of E versus k is proportional to the group velocity

(b) the slope of E versus k has its maximum value at

(d) the slope of E versus k is non-zero for all k the interval

Ans. (a)

Sol.

55. An external magnetic field of magnitude H is applied to a Type-I superconductor at a temperature below the transition point. Then which one of the following statements is NOT true for H less than the critical field H_C?

(a) the sample is diamagnetic

(b) it magnetization varies linearly with H

(d) the sample exhibits mixed states of magnetization near H_C

Ans. (d)

Sol. Below H_C, the sample behaves as a diamagnetic. Its magnetization varies linearly with H (slope = –1) and Meissner effect will be followed by the superconductor. The statement "the sample exhibits mixed states of magnetization near H_C" is not true for type-I superconductor. It is true for type-II superconductors.

56. A ferromagnetic material has a Curie temperature 100K. Then

(a) its susceptibility is doubled when it is cooled from 300K to 200K

(b) all the atomic magnets in it get oriented in the same direction above 100K

(d) the plot of its susceptibility versus temperature is linear with an intercept T_C

Ans. (a)

Sol. The magnetic susceptibility of a ferromagnetic material is given from Curie-Weiss equations

Above T_C, atomic magnets will be randomly oriented and the material will become paramagnetic.

From equations (ii) and (iii), we get .

57. The point group symmetrics of the three molecules shown in Figs. 1–3 are respectively

(a) C_2h, C_2v, C_2h

(b) C_2v, C_2h, C_2h

(d) C_2v, D_2h, C_2h

Ans. (c)

Sol.

58. The energy density of states of an electron in a one dimensional potential well of infinitely high walls is (the symbols have their usual meaning)

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Energy eigenvalues of an electron in a one dimensional potential well of infinitely high walls is

59. Which one of the following statements concerning the Compton effect is NOT correct?

(a) The wavelength of the scattered photon is greater than or equal to the wavelength of the incident photon

(b) The electron can acquire a kinetic energy equal to the energy of the incident photon

(c) The energy of the incident photon equals to the kinetic energy of the electron plus the energy of the scattered photon

(d) The kinetic energy acquired by the electron in the largest when the incident and scattered photons move in opposite directions

Ans. (b)

Sol. In Compton effect experiment, the relation between the wavelength of the scattered photon and wavelength of the incident photon are related as

Thus, kinetic energy will be largest if i.e. incident and scattered photons move in the opposite direction.

If the electron acquired a kinetic energy equal to energy of the incident photon, then energy and momentum conservation cannot be satisfied simultaneously.

60. If the photon were to have a finite mass, then the Coulomb potential between two stationary charges separated by a distance r would

(a) be strictly zero beyond some distance

(b) fall off exponentially for large values of r

(d) fall off as 1/r for large values of r

Ans. (b)

Sol. If the photon has finite mass, then form of Coulomb potential will be replaced by yukawa potential

61. A stationary particle in free space is observed to spontaneously decay into two photons. This implies that

(a) the particle carries electric charge

(b) the spin of the particle must be greater than or equal to 2

(d) the mass of the particle must be greater than or equal to the mass of the hydrogen atom

Ans. (c)

Sol. Since, photons are bosons having integral spins. Therefore, the particle will be a boson.

62. The masses of a hydrogen atom, neutron and ²³⁸U₉₂ are given by 1.0078, 1.0087 and 238.0508 respectively. The binding energy of ²³⁸U₉₂ is therefore approximately equal to (taking 1 a.m.u. = 931.64 MeV)

(a) 120 MeV

(b) 1500 MeV

(d) 1800 MeV

Ans. (d)

Sol. Binding energy is given as

BE = [mass of constituents – mass of atom]c²

= [92 × 1.0078 + (238 – 92) × 1.0087 – 238.0508]c²

= 1804.59 MeV [use 1 amu = 931.64 MeV]

63. A bistable multivibrator with a saturation voltage +5V is shown in the diagram. The positive and negative threshold at the inverting terminal for which the multivibrator will switch to the other state are

(a) +5/11V

(b) +10/11V

(d) +11V

Ans. (a)

Sol. Given circuit is positive feedback "Schmitt trigger"

Output of positive feedback circuit is V₀ = ± V_SAT

64. An avalanche effect is observed in a diode when

(a) the forward voltage is less than the breakdown voltage

(b) the forward voltage exceeds the breakdown voltage

(d) the diode is heavily doped and forward biased

Ans. (c)

Sol.

65. Which of the given relations between the Boolean variables P and Q is NOT correct? (In the notation used here, P' denotes NOT P and Q' denotes NOT Q)

(a) PQ' + PQ = P

(b) (PQ)' + P' + Q'

(d) PQ' + Q = P

Ans. (d)

Sol.

Data for Q. No. 66 to 67

Consider the vector V = r/r³

66. The surface integral of this vector over the surface of a cube of size a and centered at the origin

(a) 0

(b)

(c)

(d)

Ans. (d)

Sol.

67. Which one of the following is NOT correct?

(a) Value of the line integral of this vector around any closed curve is zero

(b) This vector can be written as the gradient of some scalar function

(d) This vector can represent the magnetic field of some current distribution

Ans. (d)

Sol.

But, magnetic field is not conservative.

Data for Q. No. 68 to 69

Consider the motion of a particle in the potential V(x) shown in the figure.

68. Suppose the particle has a total energy E = V₁ in the figure. Then the speed of the particle is zero when it is at

(a) point P

(b) point Q

(d) point T

Ans. (a)

Sol. Energy line intersects potential at P, Therefoe P speed is zero.

69. Which one of the following statements is NOT correct about the particle?

(a) It experience no force when its position corresponds to the point Q on the curve

(b) It experience no force when its position corresponds to the point R on the curve

(d) It will be in a closed orbit between P and R if E < V₁

Ans. (c)

Sol. At S potential energy is not minimum therefore speed cannot be maximum at this point.

Data for Q. No. 70 to 71

A particle of mass m moving with speed v collides with a stationary particle of equal mass. After the collision, both the particles move. Let be the angle between the two velocity vectors

70. If the collision is elastic, then

(a) is always less than 90º

(b) is always equal to 90º

(d) cannot be deduced from the given data

Ans. (b)

Sol.

using conservation of momentum we get

square and add to get

If collision is elastic

from equation (iii) we get

i.e. two particles move at 90º to each other.

71. If the collision is inelastic, then

(a) is always less than 90º

(b) is always equal to 90º

(d) could assume any value in the range 0º to 180º

Ans. (a)

Sol. If collision is inelastic

from equation (iii) in solution (70),

Data for Q. No. 72 to 73

Consider two conducting plates of infinite extent, one plate at z = 0 and the other at z = L, both parallel to the xy plane. The vector and scalar potential in the region between the plates is given by

72. For this to represent a standing wave in the empty region between the plates

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

For standing wave, the boundary condition is

73. The energy density at z = 0 and t = 0 is

(a) 0

(b)

(c)

(d)

Ans. (a)

Sol. The energy density,

At t = 0 and z = 0, u = 0

Data for Q. No. 74 to 75

A particle is located in a three dimensional cubic well of width L with impenetrable walls.

74. The sum of the energies of the third and the fourth levels is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Energy eigenvalues of a particle located in three dimensional cubic well of width L with impenetrable walls, are

Sum of third level energy and fourth level energy = 2^nd excited state energy + 3^rd excited state energy =

75. The degeneracy of the fourth level is given by

(a) 1

(b) 2

(d) 4

Ans. (c)

Sol. Degeneracy of the fourth level i.e. third excited state is 3 and corresponding combinations of (n_x, n_y, n_z) are (1, 1, 3), (1, 3, 1) and (3, 1, 1).

Data for Q. No. 76 to 77

The normalized wave functions and correspond to the ground state and the first excited state of a particle in a potential. You are given the information that the operator acts on the wave functions as and .

76. The expectation value of A for the state is

(a) –0.32

(b) 0.0

(d) 0.96

Ans. (d)

Sol.

77. Which of the following are eigenfunctions of ?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Data for Q. No. 78 to 79

In the presence of an inhomogeneous weak magnetic field, spectral lines due to transitions between two sets of states were observed.

78. The types of Zeeman effect observed in (1) and (2) respectively are

(a) normal, normal

(b) anomalous, anomalous

(d) normal, anomalous

Ans. (b)

Sol. Both transitions are between multiplets. Therefore, both are anomalous Zeeman effect.

79. The number of levels into which each of the above four terms split into respectively is

(a) 6, 4, 10, 8

(b) 4, 6, 10, 12

(d) 9, 5, 12, 10

Ans. (c)

Sol. In anomolous Zeeman effect, each J level splits in 2J + 1 levels. Here, J values are 5, 4, and . Therefore, 2J + 1 values are equal to 11, 9, 6 and 4.

Data for Q. No. 80 to 82

A system consists of three spin-half particles, the z components of whose spins S_z(1), S_z(2) and S_z(3) can take value +1/2 and –1/2. The total spin of the system is S_z = S_z(1) + S_z(2) + S_z(3).

80. The total number of possible micro-states of this system is

(a) 3

(b) 6

(d) 8

Ans. (d)

Sol. Number of possible micro-states for a collection of N particles having spin quantum numbers

81. The total number of micro-states with S_z = 1/2 is

(a) 3

(b) 5

(d) 7

Ans. (a)

Sol. For , only the following combinations of S_z(1), S_z(2), S_z(3) are possible.

So the total number of possible microstates are 3.

82. Consider an ensemble of systems where each microstate has equal probability. The ensemble average of S_z is

(a) –1/2

(b) 0

(d) 3/2

Ans. (b)

Sol. The following microsates are possible:

Since each state is equally probable, only above configurations are possible. Hence, ensemble average

Data for Q. No. 83 to 84

A gas of N particles is enclosed in a volume V at a temperature T. The logarithm of the partition function is given by ln Z = N ln {(V – bN)(kBT)^3/2} where b is a constant with appropriate dimensions

83. If P is the pressure of the gas, the equation of state is given by

(a) P(V – bN) = Nk_BT

(b) P(v – bN) = k_BT

(d) P(V – b) = Nk_BT

Ans. (a)

Sol.

The Helmholtz free energy is

The pressure is given by

84. The interval energy of the gas is given by

(a) U = (1/2)k_BT

(b) U = Nk_BT

(d) U = 2Nk_BT

Ans. (c)

Sol. The internal energy is

Data for Q. No. 85 to 86

A crystal belongs to a face centered cubic lattice with four atoms in the unit cell. The size of the crystal is 1 cm and its unit cell dimension is 1 nm. f is the scattering factor of the atom.

85. The number of atoms in the crystal is

(a) 2 × 10²¹

(b) 4 × 10²¹

(d) 4 × 10²⁴

Ans. (b)

Sol. Unit cell dimension (a) = 1 nm = 1 × 10^–9 m and size of the crystal = 1 cm.

Number of unit cells in the crystal

Number of atoms in the crystal = (number of atoms/unit cell) × number of unit cells = 4 × 10²¹ atoms.

86. The structure factors for (0 1 0) and (2 0 0) reflections respectively are

(a) 2f and zero

(b) zero and 4f

(d) zero and zero

Ans. (b)

Sol. The amplitude of diffracted X-ray beam for fcc lattice having four atoms at (0, 0, 0), and positions is given by

It is obvious that the amplitude of diffracted X-ray beam is non-zero only if h, k and l are all even or all odd and has a value equal to 4f. The amplitude of diffracted X-ray beam vanishes for all other odd-even combinations of h, k and l.

Hence, for (010):

Data for Q. No. 87 to 88

An atomic bomb consisting of ²³⁵U explodes and releases an energy of 10¹⁴J. It is known that each ²³⁵U which undergoes fission releases 3 neutrons and about 200 MeV of energy. Further, only 20% of the ²³⁵U atoms in the bomb undergo fission.

87. The total number of neutrons released is about

(a) 4.7 × 10²⁴

(b) 9.7 × 10²⁴

(d) 3.7 × 10²⁵

Ans. (b)

Sol. Number of fissions (or reactions or ²³⁵U undergoing fission)

Number of neutrons released = 3 × number of fissions

88. The mass of ²³⁵U in the bomb is about

(a) 1.5 kg

(b) 3.0 kg

(d) 12 kg

Ans. (c)

Sol. Let the total mass f ²³⁵U is m kg.

Therefore, the number of nuclei in m kg mass are

According to question, each ²³⁵U gives 200 MeV but 20% of ²³⁵U nuclei in the bomb undergoes fission. Hence, the total energy produced after the fission of m mass atomic bomb

Data for Q. No. 89 to 90

The circuit below represents a non-inverting integrator

89. For high frequencies the input impedance is

(a) 0

(b) R

(c)

(d)

Ans. (b)

Sol. An ideal op-amp has infinite input resistance. So, there will be no current passing through op-amp. So, V_a and V_b ill be virtual short i.e. V_a = V_b.

V₁ = i₁·z