PYQ 2004 - VedPrep

GATE PHYSICS 2004
Previous Year Question Paper with Solution.

1. For the function , the value of at = x = y = 1 is

(a) 5

(b)

(d)

Ans. (d)

Sol.

2. The average of the function f(x) = sin x in the interval is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

3. Identify the points of unstable equilibrium for the potential shown in the figure.

(a) p and s

(b) q and t

(d) r and s

Ans. (c)

Sol.

and at unstable equilibrium there is local maxima

4. Which one of the following remains invariant under Lorentz transformations ?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

is invariant under Lorentz transformation as shown below.

According to Lorentz transformation

5. A charge +q is kept at a distance of 2R from the centre of a grounded conducting sphere of radius R. The image charge and its distance from the centre are, respectively

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The image charge,

And distance from the centre

6. The state of polarization of light with the electric field vector is

(a) linearly polarized along z-direction

(b) linearly polarized at –45º to x-axis

(d) ellliptically polarized with the major axis along x-axis

Ans. (b)

Sol.

Therefore, it is a linearly polarized light at –45° to x-axis.

7. The resonance widths of particle resonances satisfy the relation . Their life-times satisfy the relation

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

8. The time-independent Schrödinger equation of a system represents the conservation of the

(a) total binding energy of the system

(b) total potential energy of the system

(d) total energy of the system

Ans. (d)

Sol. Correct option is (d)

9. In a hydrogen atom, the accidental or Coulomb degeneracy for the n = 4 state is

(a) 4

(b) 16

(d) 32

Ans. (b)

Sol. Coulomb degeneracy of n^th state of hydrogen atom is g_n = n². For n = 4 state, g₄ = 16

10. The Hamiltonian of a particle is given by is angular momentum. The Hamiltonian does NOT commute with

(a)

(b)

(d)

Ans. (c)

Sol.

11. The spectral terms for a certain electronic configuration are given by ³D, ¹D, ³P, ¹P, ⁵S, S³. The term with the lowest energy is

(a) ⁵S

(b) ³P

(d) ³S

Ans. (a)

Sol. Term with largest multiplicity (2S + 1) will be ground state. ⁵S term has largest multiplicity equal to 5.

12. The degeneracy of the spectral term ³Fis

(a) 7

(b) 9

(d) 21

Ans. (d)

Sol.

Therefore, total angular momentum J = (l + s) to |l – s| = (3 + 1) to (3 – 1) = 4, 3, 2.

Therefore, degeneracy of special term ²F is 21.

13. The Lande g factor for the level ³D₃ is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

14. All vibrations producing a change in the electric dipole moment of a molecule yield

(a) Raman spectra

(b) Infrared spectra

(d) X-ray spectra

Ans. (b)

Sol. Infrared spectra will be observed when there is a change in the electric dipole moment of a molecule due to vibration.

15. For any process, the second law of thermodynamics requires that the change of entropy of the universe be

(a) positive only

(b) positive or zero

(d) negative or zero

Ans. (b)

Sol. The second law of thermodynamics states that the entropy of the universe always increases in a irreversible process and remains constant in a reversible process, i.e., dS > 0.

16. The dimension of phase space of ten rigid diatomic molecules is

(a) 5

(b) 10

(d) 100

Ans. (d)

Sol. The degree of freedom of each rigid diatomic molecules, f = (3N – C) = 3 × 2 – 1 = 5

So, the phase space dimension of each diatomic molecules = 2f = 10

So, the phase space dimension of 10 diatomic molecules = 100

17. The specific heat of an ideal Fermi gas in 3-dimension at very low temperatures (T) varies as

(a) T

(b) T^3/2

(d) T³

Ans. (a)

Sol. The specific heat of Fermi gas is given by

18. Which one of the following is a first order phase transition ?

(a) Vaporization of a liquid at its boiling point

(b) Ferromagnetic to paramagnetic

(d) Superconducting to normal state

Ans. (a)

Sol. The first order phase transition involves the concept of latent heat. Moreover, the vaporization latent heat of vaporization and first derivative of Gibbs free energy is discontinuous.

19. The c/a ratio for an ideal hexagonal closed packed structure is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Let a is interatomic distance within a layer and c is height of the unit cell

Unit cell of hexagonal closed packed structure can be made as:

If the centre of spheres are in contact, then

20. The number of independent elastic constants in an isotropic cubic solid is

(a) 1

(b) 2

(d) 4

Ans. (c)

Sol. Youngs modulus, bulk modulus and Poisson's ratio are the independent elastic constants in an isotropic cubic solid.

21. The effective mass of an electron in a semiconductor

(a) can never be positive

(b) can never be negative

(d) depends on its spin

Ans. (c)

Sol.

22. The critical magnetic field for a solid in superconducting state

(a) does not depend upon temperature

(b) increases if the temperature increases

(d) does not depend on the transition temperature

Ans.

Sol. , where H_C(0) is critical field at 0 K. Now, if T increases, T² will also increase.

Thus, H_C(T) will decrease and vice-versa.

23. The volume of a nucleus in an atom is proportional to the

(a) mass number

(b) proton number

(d) electron number

Ans. (a)

Sol.

24. As one moves along the of stability from ⁵⁶Fe to ²³⁵U nucleus, the nuclear binding energy per particle decreases from about 8.8 MeV to 7.6 MeV. This trend is mainly due to the

(a) short range nature of the nuclear forces

(b) long range nature of the Coulomb forces

(d) spin dependence of the nuclear forces

Ans. (b)

Sol. The decrease of binding energy from ⁵⁶Fe to ²³⁵U nuclear is due to the long range nature of Coulomb forces.

25. A thermal neutron having speed v impinges on a ²³⁵U nucleus. The reaction cross-section is proportional to

(a) v^–1

(b) v

(d) v^–1/2

Ans. (a)

Sol. The reaction cross-section is given by

where, R is the reaction rate density

n is the number density of the particles in the neutron beam.

v is the velocity of neutron

N is the number of density of the nuclei in the target.

26. Choose the particle with zero Baryon number from the list given below.

(a) pion

(b) neutron

(d)

Ans. (a)

Sol. Since, neutron, proton and are member of Baryon family so they will have B = 1

But pion belong to Hadrons family (meson). Therefore, it will have B = 0

27. A bipolar junction transistor with one junction forward biased and either the collector or emitter open, operates in the

(a) cut-off region

(b) saturation region

(d) active region

Ans. (c)

Sol. Considering Common Base configuration

When input junction is in forward bias, we have possible regions of operation-active or saturation. But we can't comment about any region.

Cut-off (invalid)

Saturation (no comment about output junction)

Active (no comment about output junction)

Pinch-off (doesn't exist in BJT characteristics)

Correct option is (c), Question is ambiguous

28. A field effect transistor is a

(a) unipolar device

(b) special type of biopolar junction transistor

(d) device with low input impedance

Ans. (a)

Sol. FET is unipolar because it has only majority carriers.

FET is high impedance device

FET is having low gain

FET is having low gain]

FET "VCCS" voltage control current source.

29. The inverting input terminal of an operational amplifier (op-amp) is shorted with the output terminal apart from being grounded. A voltage signal v_i is applied to the non-inverting input terminal of the op-amp. Under this configuration, the op-amp functions as

(a) an open loop inverter

(b) a voltage to current converter

(d) an oscillator

Ans. (c)

Sol. Drawing circuit as per statement

Note: Used for impedance matching.

30. A half-adder is a digital circuit with

(a) three inputs and one output

(b) three inputs and two outputs

(d) two inputs and two outputs

Ans. (d)

Sol.

Hence, it has 2 inputs and 2 outputs (Sum, Carry)

Note: Full adder has 3 inputs and 2 outputs.

31. A real traceless 4 × 4 unitary matrix has two eigenvalues –1 and +1. The other eigenvalues are

(a) zero and +2

(b) zero and +1

(d) –1 and +1

Ans. (d)

Sol. Sum of eigenvalues = Trace of the marix

32. The eigenvalues of the matrix are

(a) +1 and +1

(b) zero and +1

(d) –1 and +1

Ans. (c)

Sol.

33. The inverse of the complex number is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

34. The value of , where C is a unit circle (anti-clockwise) centered at the origin in the complex z-plane is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

35. The Laplace transform . Therefore, the Laplace transform of

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

36. A periodic function has the Fourier series representation

. Using this, one finds the sum to be

(a) 2 ln 2

(b)

(c)

(d)

Ans. (b)

Sol.

Using, Parseval's formula, we get

37. The Fourier transform F(k) of a function f(x) is defined as . The F(k) for f(x) = exp(–x²) is [Given : ]

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

38. The Lagrangian of a particle moving in a plane under the influence of a central potential is given by

The generalized momenta corresponding to r and are given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

39. A particle of mass m is attached to a thin uniform rod of length a and mass 4m. The distance of the particle from the center of mass of the rod is a/4. The moment of inertia of the combination about an axis passing through O normal to the rod is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. I₀ = I_rod + I_particle

40. A rigid frictionless rod rotates anticlockwise in a vertical plane with angular velocity . A bead of mass m moves outward along the rod with constant velocity . The bead will experience a coriolis force

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

41. The Hamiltonian corresponding to the Langrangian is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

42. The value of the Poisson bracket are constant vectors, is

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

43. A mass m is connected on either side with a spring each of spring constant k₁ and k₂. The free ends of springs are tied to rigid supports. The displacement of the mass is x from equilibrium position. Which one of the following is true ?

(a) The force acting on the mass is –(k₁k₂)^1/2x

(b) The angular momentum of the mass is zero about the equilibrium point and its Lagrangian is

(d) The angular momentum of the mass is and the Lagrangian of the system is

Ans. (b)

Sol.

at ay instant are along same line. Therefore

44. An electron gains energy so that its mass becomes 2m₀. Its speed is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

45. A conducting sphere of radius R has charge +Q on its surface. If the charge on the sphere is doubled and its radius is halved, the energy associated with the electric field will

(a) increase four times

(b) increase eight times

(d) decrease four times

Ans. (b)

Sol.

46. A conducting sphere of radius R is placed in a uniform electric field directed along +z axis. The electric potential for outside points is given as , where r is the distance from the centre and is the polar angle. The charge density on the surface of the sphere is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

47. A circular arc QTS is kept in an external magnetic field as shown in figure. The arc carries a current I. The magnetic field is directed normal and into the page. The force acting on the arc is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The force on the elementary length

48. A plane electromagnetic wave of frequency is incident on an air-dielectric interface. The dielectric is linear, isotropic, non-magnetic and its refractive index is n. The reflectance (R) and transmittance (T) from the interface are

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

49. The electric field of a plane e.m. wave is . If are cartesian unit vectors, the wave vector of the e.m. wave is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Comparing this with plane wave equation

50. The dispersion relation for a low density plasma is , where w₀ is the plasma frequency and c is the speed of light in free space. The relationship between the group velocity (v_g) and phase velocity (v_p) is

(a) v_p = v_g

(b)

(d)

Ans. (c)

Sol.

51. A Michelson interferometer is illuminated with monochromatic light. When one of the mirrors is moved through a distance of 25.3 µm, 92 fringes pass through the cross-wire. The wavelength of the monochromatic light is

(a) 500 nm

(b) 550 nm

(d) 650 nm

Ans. (a)

Sol. For the m^th order bright fring

52. A beam of mono-energetic particles having speed v is described by the wave function , where u(x) is a real function. This corresponds to a current density

(a) u²(x)v

(b) v

(d) u²(x)

Ans. (a)

Sol.

53. The wave function of a spin-less particle of mass m in a one-dimensional potential V(x) is corresponding to an eigenvalue . The potential V(x) is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. According to Schrodinger equation,

54. Two spin interact via a potential . The contribution of this potential in the singlet and triplet states, respectively, are

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

55. The wave function of a one-dimensional harmonic oscillator is for the ground state . In the presence of a perturbing potential of , the first order change in the ground state energy is

(a)

(b) (3E₀)10^–4

(c)

(d) (E₀)10^–4

Ans. (c)

Sol.

Applying normilazation condition,

So, first order correction to energy,

56. The L, S and J quantum numbers corresponding to the ground state electronic configuration of Boron (Z = 5) are

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Atomic number boron, Z = 5. The electronic configuration of boron is given by 1s², 2s₂, 2p¹ i.e., there is one unpaired electron in 2p subshell.

According to L-S couplnig, J will have values from |l + s| to |l – s|

For less than half filled orbital, J value coresponds to ground state.

57. The degeneracies of the J-states arising from the ³P term with spin-orbit interaction are

(a) 1, 3, 5

(b) 1, 2, 3

(d) 2, 6, 10

Ans. (a)

Sol.

Now, J = |1 + 1| to |1 –1| = 2, 1, 0.

So, the term is ³p_{0, 1, 2} i.e., ³p₀, ³p₃ and ³p₂

Now, degeneracy of individual terms are given by

58. Assuming that the L-S coupling scheme is valid, the number of permitted transitions from ²P_3/2 to ²S_1/2 due to a weak magnetic field is

(a) 2

(b) 4

(d) 10

Ans. (c)

Sol.

According to selection rule, , total number of permitted transitions from ²P_3/2 to ²S_1/2 due to a weak magnetic field is 6.

59. Consider the pure rotational spectrum of a diatomic rigid rotor. The separation between two consecutive lines in the spectrum

(a) is directly proportional to the moment of inertial of the rotor

(b) in inversely proportional to the moment of inertia of the rotor

(d) is directly proportional to the square of the interatomic separation

Ans. (b)

Sol. For pure rational spectrum of a diatomic rigid rotar, the separation between two consecutive lines in the spectrum is given by

where, I is moment of inertia.

Thus, the seperation inversely proportional to the momoent of inretia of the roto.

60. Light of wavelength 1.5 µm incident on a material with a characteristic Raman frequency of 20 × 10¹² Hz results in a Stokes-shifted line of wavelength

[Given : c = 3 × 10⁸ m.s^–1]

(a) 1.47 µm

(b) 1.57 µm

(d) 1.77 µm

Ans. (c)

Sol. The frequency corresponds waavelength 1.5 µm incident on material is

The frequency of Raman is V_Raman = 20 × 10¹² Hz

This is Raman shift = 20 × 10¹² Hz

Thus, frequency of Stocks line is

v = v – = 2 × 10¹⁴ – 20 × 10¹² Hz = 180 ×10¹² Hz

Thus the wavelength of Stocks line is given by

61. Consider black body radiation in a cavity maintained at 2000 K. If the volume of the cavity is reversibly and adiabatically increased from 10 cm³ to 640 cm³, the temperature of the cavity changes to

(a) 800 K

(b) 700 K

(d) 500 K

Ans. (d)

Sol. According to Wien's displacement law of blackbody radiation, TV1/3 = constant.

where T₁ = 2000 K, V₁ = 10 cm³, T₂ = ?, V₂ = 640 cm³

62. The equation of state of a dilute gas at very high temperature is described by , where v is the volume per particle and B(T) is a negative quantity. One can conclude that this is a property of

(a) a van der Waals gas

(b) an ideal Fermi gas

(d) an ideal inert gas

Ans. (a)

Sol.

Clearly, B(T) can be negative for Van der Waals' gas only, i.e., B(T) < 0

63. In the region of co-existence of a liquid and vapor phases of a material

(a) C_p and C_v are both infinite

(b)

(c)

(d)

Ans. (a)

Sol. The region of co-existence of a liquid and a vapour is a region order phase transition and the specific heats C_p and C_V are both infinite.

64. A doped Germanium crystal of length 2 cm, breadth 1 cm and width 1 cm, carries a current of 1 mA along its length parallel to +x-axis. A magnetic field of 0.5 T is applied along +z-axis. Hall voltage of 6 mV is measured with negative polarity at plane. The sign and concentration of the majority charge carrier are, respectively. [Given : e = 1.6 × 10^–19 C]

(a) positive and 5.2 × 10¹⁹ m^–3

(b) negative and 5.2 × 10¹⁹ m^–3

(d) negative and 10.4 × 10¹⁹ m^–3

Ans. (a)

Sol. Hall voltage,

and this is n-type

65. The temperature dependence of the electrical conductivity of two intrinsic semiconductors A and B is shown in the figure. If E_A and E_B are the band gaps of A and B respectively, which one of the following is true ?

(a) E_A > E_B

(b) E_A < E_B

(d) E_A and E_B both depend on temperature

Ans. (a)

Sol.

Now, since refractive index, n = 1.5

66. If the static dielectric constant of NaCl crystal is 5.6 and its optical refractive index is 1.5, the ratio of its electric polarizability to its total polarizability is

(a) 0.5

(b) 0.7

(d) 0.9

Ans.

Sol.

67. Which one of the following statements is TRUE ?

(a) Magnetic tapes are made of Iron

(b) Permanent magnets are made from ferrites

(d) Optoelectronic devices are made from soft ferrites

Ans. (b)

Sol. Parmanent magnetic made from hard ferromagnetic materials.

68. Which one of the following statements is not true?

(a) Entropy decreases markedly on cooling a superconductor below the critical temperature, T_c

(b) The electronic contribution to the heat capacity in the superconducting state has an exponential form with an argument proportional to T ^–1, suggestive of an energy gap

(d) Critical temperature of superconductors does not vary with the isotopic mass

Ans. (d)

Sol. From option (a) enthalpy always decreaess on cooling a superconductor below its critical temperatur. This inicates that superconducting state is more ordere than normal state.

From option (b) electronic contribution to heat capacity also has exponential form.

From option (c) A type-I superconductor is also perfet diamagnet

From option (d) Critical temperature vary with isotopic mass and this is known as isotopic effect.

69. The form factor of Rutherford scattering is obtained by choosing a delta function for the charge density . The value of the form factor is

(a) unity

(b) infinity

(d) undefined

Ans. (a)

Sol.

70. Deuteron in its ground state has a total angular momentum J = 1 and a positive parity. The corresponding orbital angular momentum L and spin S combinations are

(a) L = 0, S = 1 and L = 2, S = 0

(b) L = 0, S = 1 and L = 1, S = 1

(d) L = 1, S = 1 and L = 2, S = 1

Ans. (c)

Sol. In ground state deuteron remains 96% of time in ³S state for which (L = 0, S = 1) and 4% of time in 3D state for which (L = 2, S =1)

71. Which one of the following reaction is allowed ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. (a) here Lepton number and angular momentumis not conserved. Hence reaction is not allowed.

(b) Baryon number and angular momentum is not conserved. Hence, reaction not allowed.

(d)

72. What should be the values of the components R and R₂ such that the frequency of the Wien Bride oscillator is 300 Hz ?

[Given : ]

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given circuit is standard Wein Bridge Oscillator. The condition for sustaining oscillation in the Wien bridge oscillator is gain A > 3.

frequency can be calculated by finding feedback factor and equating it's imiginary part to zero

73. Figure shows a common emitter amplifier with . What is the maximum peak to peak input signal (v_s) for which is distortion-free output may be obtained ?

[Assume V_BE = 0 and ]

(a) 40 mV

(b) 60 mV

(d) 100 mV

Ans. (d)

Sol.

Now, the input should be such that the transistor does not go into saturation region.

Thus, theoritically an input singal of 100 mV can be applied.

74. Calculate the collector voltage (v_c) of the transistor circuit is shown in the figure.

[Given : ]

(a) 3.78 V

(b) 3.82 V

(d) 9.7 V

Ans. (a)

Sol.

By putting value I_C = 24I_B + (25 × 20µA)

I_C = 24I_B + (500 µA) …(i)

Apply KVL at collector-base junction

10 = 100I_B + 0.3

I_B = 0.097 mA

Put the value of I_B in equation (1)

I_C = (24 × s0.097 mA) + (0.5 mA)

I_C = (2.328 + 0.5) mA = 2.828 mA

Apply KVL at output

10 = 1_C × 2.2 + V_C

V_C =3.778 Volt

75. Figure shows a practical integrator with . If a step (dc) voltage of +3 V is applied as input for 0 < t < 4 (t is in seconds), the output voltage is

(a) a ramp function of –6 V

(b) a step function of –12 V

(d) a ramp function of –4 V

Ans. (d)

Sol. 3V is applied ofr 0 < t < 4 sec.

Put the given value

By applying Inverse Lapace Transforms

It is Ramp function.

76. The Boolean expression reduces to

(a)

(b) D

(c)

(d)

Ans. (d)

Sol.

Common Data for Q. 77 and Q. 78

Consider the differential equation y" + p(x)y' + q(x)y(x) = 0.

77. If xp(x) and x²q(x) have the Taylor series expansions

xp(x) = 4 + x + x² + ....

x²q(x) = 2 + 3x + 5x² + ....

then the roots of the incidicial equation are

(a) –1, 0

(b) –1, –2

(d) –1, 2

Ans. (b)

Sol. Correction option is (b)

78. If p(x) = 0 with the Wronskian at x = 0 as W(x = 0) = 1 and one of the solutions is x, then the other linearly independent solution which vanishes at x = 1/2 is

(a) 1

(b) 1 – 4x²

(d) –1 + 2x

Ans. (d)

Sol. Correct option is (d)

Common Data for Q. 79 and Q. 80

Consider a comet of mass m moving in a parabolic orbit around the Sun. The closest distance between the comet and the Sun is b, the mass of the Sun is M and the universal gravitation constant is G.

79. The angular momentum of the comet is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Energy of connect in parabolic orbit is zero.

E = 0

near the sun

angular momentum

80. Which one of the following is TRUE for the above system ?

(a) The acceleration of the comet is maximum when it is closest to the Sun

(b) The linear momentum of the comet is a constant

(d) The kinetic energy of the comet is a constant

Ans. (a)

Sol. Nearest the Sun, force is maximum, therefore, acceleration is maximum

Common Data for Q. 81 and Q. 82

Let are cartesian unit vectors, represent an electric field of plane electro magnetic wave of frequency .

81. Which one of the following statements is TRUE ?

(a) The magnitude of the electric field is attenuated as the wave propagates

(b) The energy of the e.m. wave flows along the x-direction

(d) The speed of the wave is the same as c (speed of light in free space)

Ans. (a)

Sol.

This says that the amplitude decays as the wave propagates.

82. The magnetic field of the wave is

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

We have,

Common Data for Q. 83 and Q. 84

A particle is confined to the region 0 < x < L in one dimension

83. If the particle is in the first excited state, then the probability of finding the particle is maximum at

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

The probability density in the first excited state, is

84. If the particle is in the lowest energy state, then the probability of finding the particle in the region is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. The wave function of the lowest energy stage, is

Common Data for Q. 85 and Q. 86

The one-electron states for non-interacting electrons confined in a cubic box of side a are etc.

85. The energy of the lowest state is

(a) zero

(b)

(c)

(d)

Ans. (d)

Sol. The energy of the electron in n^th state of the box is given by

The energy of the lowest state (n_x = 1, n_y = 1, n_z = 1)

86. The degeneracy (including spin) of the level is

(a) 2

(b) 4

(d) 8

Ans. (c)

Sol.

Possible combination of (n_x, n_y, n_z) are (3, 1, 1), (1, 3, 1), (1, 1, 3)

So, the degeneraacy (including spin) of the level = 3 × 2 = 6

Common Data for Q. 87 and Q. 88

An ensemble of N three level systems with energies is in thermal equilibrium at temperature T. Let .

87. If , the probability of finding the system in the level is

(a)

(b) (cos h 2)^–1

(d) (1 + 2 cosh 2)^–1

Ans. (d)

Sol.

88. The free energy of the system at high temperature is approximately

(a) –Nk_BTx²

(b) –Nk_BT[ln 2 + x²/2]

(d) –Nk_BT ln 3

Ans. (d)

Sol. The free energy is

Common Data for Q. 89 and Q. 90

The nucleus ⁴¹Ca can be described by the single particle shell model.

89. The single particle states occupied by the last proton and the last neutron, respectively, are given by

(a) d_5/2 and f_7/2

(b) d_3/2 and f_5/2

(d) d_3/2 and f_7/2

Ans. (d)

Sol.

90. The ground state angular momentum and parity of ⁴¹Ca are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. N = 21, last neutron (¹f_7/2)¹

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Previous Year Question Paper with Solution.

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Previous Year Question Paper with Solution.