PYQ 2010 - VedPrep

GATE PHYSICS 2010
Previous Year Question Paper with Solution.

1. Consider an anti-symmetric tensor P₀ with the indices i and j running from 1 to 5. The number of independent components of the tensor is

(a) 3

(b) 10

(d) 6

Ans. (b)

Sol. The number of independent component of a antisymmetric tensor P_ij of rank`2', is given by (N² – N) Here, the indices i and j are running from 1 to 5. So, N = 5

Therefore, number of independent components

2. The value of the integral , where the contour C is the unit circle |z – 2| = 1, is

(a)

(b)

(c)

(d) 0

Ans. (d)

Sol.

3. The eigenvalues of the matrix are

(a) 5, 2, –2

(b) –5, –1, –1

(d) –5, 1, 1

Ans. (c)

Sol.

Only option (c) satisfy both relations.

4. If , then the Laplace transform of f(x) is

(a) s^–2e^3s

(b) s²e^–3s

(d) s^–2e^–3s

Ans. (d)

Sol.

5. The valence electrons do not directly determine the following property of a metal.

(a) Electrical conductivity

(b) Thermal conductivity

(d) Metallic lustre

Ans. (c)

Sol. Electrical conductivity, thermal conductivity and metallic lustre properties are valence electron dependent while shear modulus is independent of valence electrons.

6. Consider X-ray diffraction from a crystal with a face-centered-cubic (fcc) lattice. The lattice plane for which there is NO diffraction peak is

(a) (2, 1, 2)

(b) (1, 1, 1)

(d) (3, 1, 1)

Ans. (a)

Sol. For fcc lattice, the amplitude of diffiaceted X-ray beam is given by

7. The Hall coefficient, R_H of sodium depends on

(b) The effective charge carrier mass and carrier density

(b) The charge carrier density and relaxation time

(d) The effective charge carrier mass

Ans. (c)

Sol.

8. The Bloch theorem states that within a crystal, the wave function , of an electron has the form

(a) is an arbitrary function and is an arbitrary vector.

(b) is an arbitrary function and is a reciprocal lattice vector.

(d) is a lattice vector and is an arbitrary vector.

Ans. (d)

Sol. Option (d) is the correct statement of Bloch theorem.

9. In an experiment involving a ferromagnetic medium, the following observations were made. Which one of the plots does NOT correctly represent the property of the medium ?

(T_C is the Curie temperature)

(a)

(b)

(c)

(d)

Ans. (d)

Sol. For ferromagnetic substance, the specific heat is maximum at T_c. Thus correct graph of specific heat versus T is expressed as: , where; M = Nµ tan h

Thus, option (d) is not correctly shown.

10. The thermal conductivity of a given material reduces, when it undergoes a transition form its normal state to the superconducting state. The reason is

(a) The cooper pairs cannot transfer energy to the lattice.

(b) upon the formation of cooper pairs, the lattice becomes less efficient in heat transfer.

(c) The electrons in the normal state loose their ability to transfer heat because of their coupling to the cooper pairs.

(d) The heat capacity increases on transition to the superconducting state leading to a reduction in thermal conductivity.

Ans. (a)

Sol. The thermal conductivity reduces in the transition from the normal state to the superconducting state because Cooper pairs have very less energy (10^–3 eV). They can not transfer energy to the lattice.

11. The basic process underlying the neutron -decay is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. In -decay process one neutron decay into one proton and one electron

Quark, structure of n is udd and p is tiud

Thus, reaction is written as udd

12. In the nuclear shell model the spin parity of 15N is given by

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Since, has Z = 7, N = 15 – 7 = 8

Therefore, 7Z, (¹s_1/2)² (¹p_3/2)⁴ (¹P_1/2)¹

13. Match the reactions on the left with the associated interactions on the right

(a) 1-iii, 2-ii, 3-i

(b) 1-i, 2-ii, 3-iii

(d) 1-iii, 2-i, 3-ii

Ans. (a)

Sol. (1) Lepton are produced by weak interaction. I₃ is not conserved.

(2) –ray produced through electromagnetic interaction.

(3) Baryon are produced by strong interaction. Isospin I and I₃ are conserved.

14. To detect trace amounts of a gaseous species in a mixture of gases, the preferred probing tools is

(a) ionization spectroscopy with X-rays

(b) NMR spectroscopy

(d) laser spectroscopy

Ans. (c)

Sol. For a gaseous species in a mixture of gases, the preferred probing tools is ESR spectroscopy.

15. A collection of N atoms is exposed to a strong resonant electromagnetic radiation with N_g atoms in the ground state and N_e atoms in the excited state, such that N_g + N_e = N. This collection of two-level atoms will have the following population distribution:

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Since, to establish equilibirium

16. Two state of an atom have definite parities. An electric dipole transition between these states is

(a) allowed if both the states have even parities

(b) allowed if both the states have odd parities

(d) not allowed unless a static electric field is applied

Ans. (c)

Sol. Electric dipoletransition between two states will be allowed if both the states have opposite parities. Since, selection rules are and = 0, ± 1

17. The spectrum of radiation emitted by a black body at a temperature 1000 K peaks in the

(a) visible range of frequencies

(b) infrared range of frequencies

(d) microwave range of frequencies

Ans. (b)

Sol. Using Wein's displacement law,

This wavelength lies in infrared region

18. An insulating sphere of radius a carries a charge density . The leading order term for the electric field at a distance d, far away from the charge distribution, is proportional to

(a) d^–1

(b) d^–2

(d) d^–4

Ans. (c)

Sol. From the figure we can say that total charge is zero.

But dipole moment is not zero. So efectric field will be due to dipolemoment. So, E(d) d^–3

19. The voltage resolution of a 12-bit digital to analog converter (DAC), whose output varies from –10 V to +10 V is, approximately

(a) 1 mV

(b) 5 mV

(d) 100 mV

Ans. (b)

Sol. 12–bit DAC

Range of voltage –10 to 10 = 20V.

20. In the of the following circuits, negative feedback does not operate for a negative input. Which one is it ? The opamps are running from +15 V supplies.

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

input (–)ve will make zener – ON

Since, Zener diode operates in reverse break down voltage

Diode in reverse bias for negative input, So V₀= 0

(d)

D₂ conduct for positive input. D₁ conduct for negative input.

21. A system of N non-interacting classical point particles is constrained to move on the two-dimensional surface of a sphere. The internal energy of the system is

(a)

(b)

(d)

Ans. (c)

Sol. We need two independent coordinates for each particles to know its position on the surface of the sphere. Number of degree of freedom for N particles = 2N

and energy per degree of freedom =

So internal energy ofthe system =

22. Which of the following atoms cannot exhibit Bose-Einstein condensation, even in principle ?

(a) ¹H₁

(b) ⁴He₂

(d) ³⁰K₁₉

Ans. (d)

Sol. To exhibit Bose–Einstein condensation, a particle must be Boson, i.e., it must have an integral spin. Since all protons, electrons and neutrons have halfintegral spin, only those atoms whose n_p + n_e + n_n is even will be a boson.

Since n_p + n_e + n_n for ⁴⁰K₁₉ is odd, it cannot exhibit Bose–Einstein condensation.

23. For the set of all Lorentz transformation with velocities along the x-axis, consider the two statements given below:

P : If L is a Lorentz transformation then, L^–1 is also a Lorentz transformation

Q : If are Lorentz transformation then, is necessarily a Lorentz transformation

Choose the correct option.

(a) P is true and Q is false

(b) Both P and Q are true

(d) P is false and Q is true

Ans. (b)

Sol. Lorentz transformation satisfies group properties.

Therefore both statements P and Q are correct.

24. Which of the following is an allowed wavefunction for a particle in a bound state ? N is a constant and .

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Any allowed wave function ofthe particle in bound state should satisfy the following conditions:

(1) (x) should be singled valued, finite, continuous everywhere in space,

(2) (x) should become zero at x =

= is not defined a r = 0

= N(1 – e^–ar) is not zero at r =

= is zero at x = ±

The function given in option (d) is discontinuous at r = R. Therefore, the function given is option (c) will be an allowed wave function for a particle in a bound state.

25. A particle is confined within a spherical region of radius one femtometer (10^–15 m). Its momentum can be expected to be about

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Momentum of the particle is,

26. For the complex function, , which of the following statements is correct ?

(a) z = 0 is a branch point

(b) z = 0 is a pole of order one

(d) z = 0 is an essential singularity

Ans. (c)

Sol.

So, z = 0 is a removable singular point.

27. The solution of the differential equation for , subject to the initial conditions , is

(a)

(b) –sinh (t) + tcosh (t)

(d) tsinh (t)

Ans. (d)

Sol.

The auxiallary equation is D² y – y = 0

Let the trial solution is y = e^mf

m² – 1 = 0 m = ±1

C.f. = Ae¹+ Be^–1

Total solution, y = Ae¹ + Be^–1 + t sinh t

putting the condition, y(t = 0) = 0 A + B = 0

putting the condition,

Solving above two equation, A = 0, B = 0

therefore, y = t sinh t

28. Given the recurrence relation for the Legendre polynomials

(2n + 1)x P_n(x) = (n + 1)P_{n + 1}(x) + n P_{n – 1}(x),

which of the following integrals has a non-zero value ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

29. For a two-dimensional free electron gas, the electronic density n, and the Fermi energy E_F, are related by

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The density of state is 2–D is

Since, we can have an electron with spin up or spin down in each state, we need to multiply density of state by the factor of two.

30. Far away from any of the resonance frequencies of a medium the real part of the dielectric permittivity is

(a) Always independent of frequency

(b) Monotonically decreasing with frequency

(d) A non-monotonic function of frequency

Ans. (d)

Sol. The real part of dielectric constant is given by

Therefore, Re is a non–monoatomic function of frequency

31. The ground state wavefunction of deuteron is in a superposition of s and d states. Which of the following is NOT true as a consequence ?

(a) It has a non-zero quadruple moment

(b) The neutron-proton potential is non-central

(d) The Hamiltonian does not conserve the total angular momentum

Ans. (d)

Sol. Correct option is (d).

32. The first three energy levels of ²²⁸Th₉₀ are shown below:

The expected spin-parity and energy of the next level are given by

(a) (6⁺; 400 keV)

(b) (6⁺; 300 keV)

(d) (4⁺; 300 keV)

Ans. (a)

Sol. where J is spin and I is moment of inertia

Now, for J = 2, E = 57.5 KeV

Next level is 6⁺

33. The quark content of is indicated:

In the process, , considering strong interactions only, which of the following statements is true ?

(a) The process is allowed because

(b) The process is allowed because

(d) The process is not allowed because the baryon number is violated

Ans. (c)

Sol.

34. The three principal moments of inertia of a methanol (CH₃OH) molecule have the property I_x = I_y = I and . The rotational energy eigenvalues are

(a)

(b)

(c)

(d)

Ans. (a)

Sol. CH₃OH is example of symmetric top molecules, where I_x = I_y= I and I_z

Energy level of symmetric top molecule is

35. A particle of mass m is confined in the potential

Let the wavefunction of the particle be given by , where are the eigenfunctions of the ground state and the first excited state respectively. The expectation value of the energy is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. For the given Half harmonic oscillator potential, energy eigen values of the particle are

Expectation value of the energy will be

36. Match the typical spectra of stable molecules with the corresponding wave-number range

1. Electronic spectra i. 10⁶ cm^–1 and above

2. Rotational spectra ii. 10⁵ – 10⁶ cm^–1

3. Molecular dissociation iii. 10⁰ – 10² cm^–1

(a) 1-ii, 2-i, 3-iii

(b) 1-ii, 2-iii, 3-i

(d) 1-i, 2-ii, 3-iii

Ans. (b)

Sol. Here the molecular has been compared with their corresponding wave number or we can say that energy of these spectra has been normalized in the units of cm^–1. So, in terms of energy rotational spectra correct sponds to microwave radiation which has the lowest energy. Then comes vibrational spectra (IRregion) slightly greater than both rotational and vibration.

Molecular dissociation corresponds to very high energy.

So, E_rotational <E_vibrational <E_electronic <E_{molecular dissociation}

Rotational

Molecular 10⁶ cm^–1 and above

37. Consider the operations (parity) and (time-reversal). For the electric and magnetic fields , which of the following set of transformations is correct ?

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

38. Two magnetic dipoles of magnitude m each are placed in a plane as shown.

The energy of interaction is given by

(a) zero

(b)

(c)

(d)

Ans. (d)

Sol. The interaction energy between two magnetic dipole moment is given by

39. Consider a conducting loop of radius a and total loop resistance R placed in a region with a magnetic field B thereby enclosing a flux . The loop is connected to an electronic circuit as shown, the capacitor being initially uncharged.

If the loop is pulled out of the region of the magnetic field at a constant speed u, the final output voltage V_out is independent of

(a)

(b) u

(d) C

Ans. (a)

Sol. When the loop will be pulled out of the region of the magnetic field, then amount of flux passing through the loop will gets changed. This change in flux will induced an emf in the circuit

At t = 0, loop was completly inside the magnetic field

So, = constat) (say)

Given circuit is op-amp integrator circuit

40. The figure shows a constant current source charging a capacitor that is initially uncharged.

If the switch is closed at t = 0, which of the following plots depicts correctly the output voltage of the circuit as a function of time ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Since current flow in the circuit is constant L. So, initially capacitor will start to charging

After certain time capacitor will be fully charged.

So, Voltage will not change with time.

41. For any set of inputs A and B, the following circuits give the same output Q, except one. Which one is it ?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

42. CO₂ molecule has the first few energy levels uniformly separated by approximately 2.5 meV. At a temperature of 300 K, the ratio of the number of molecules in the 4^th excited state to the number in the 2^nd excited state is about

(a) 0.5

(b) 0.6

(d) 0.9

Ans. (c)

Sol. Number of molecules in a given energy states is given by Bolt/mann law.

Ratio of molecules,

As energy is equally spaced, therefore,

43. Which among the following sets of Maxwell relations is correct ? (U-internal energy, H-enthalpy, A-Helmholtz free energy and G-Gibb's free energy)

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The enthalpy is defined as

44. For a spin-s particle, in the eigen basis of , the expectation value is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

45. A particle is placed in a region with the potential . Then,

(a) are points of stable equilibrium

(b) x = 0 is a point of stable equilibrium and is a point of unstable equilibrium

(d) There are no points of stable or unstable equilibrium

Ans. (b)

Sol.

Therefore x = 0 stable equilibrium point at x =

Therefore, x = is unstable equilibrium point.

46. A meson at rest decays into two photons, which move along the x-axis. They are both detected simultaneously after a time, t = 10s. In an inertial frame moving with a velocity V = 0.6c in the direction of one of the photons, the time interval between the two detections is

(a) 15 s

(b) 0 s

(d) 20 s

Ans. (a)

Sol. The position of two photon in the s-frame when they are detected at

x₁ – 10c m, x₂ = –10 cm

47. A particle of mass m is confined in an infinite potential well :

It is subjected to a perturbing potential within the well. Let E⁽¹⁾ and E⁽²⁾ be the corrections to the ground state energy in the first and second order in V₀, respectively. Which of the following are true ?

(a) E⁽¹⁾ = 0; E⁽²⁾ < 0

(b) E⁽¹⁾ > 0; E⁽²⁾ = 0

(d) E⁽¹⁾ < 0; E⁽²⁾ < 0

Ans. (a)

Sol. First order correction to ground state energy is,

Therefore, second order energy correction to the ground state is given by

Since, the numerator is always positive denominator is always negative

Therefore, second order correction to the ground state eitergv is always less than equal to zero i.e.

E⁽²⁾ < 0

Common Data for Questions 48 and 49 :

In the presence of a weak magnetic field, atomic hydrogen undergoes the transition:

by emission of radiation.

48. The number of distinct spectral lines that are observed in the resultant Zeeman spectrum is

(a) 2

(b) 3

(d) 6

Ans. (c)

Sol. Consider the transition, 2P_½²S_½. These terms are correspond to . In a weak magnetic field it breaks into 2J₁ + 1 = 2.A 1 = 2 components

The selection rule of transition

= 0, ± 1

Thus, in total four spectral lines are observed in transition from ²P_1/2¹S_1/2

Also, by formula, total number of transitions

49. The spectral line corresponding to the transition is observed along the direction of the applied magnetic field. The emitted electromagnetic field is

(a) Circularly polarized

(b) Linearly polarized

(d) Not emitted along the magnetic field direction

Ans. (a)

Sol. Given transition, is corresponds to = – 1

Transition = ± 1 corresponds to components. Electric field of the emitted radiation oscillates perpendicular to the applied magnetic field: Classically, this situation corresponds to two parallel dipoles oscillating with phase difference 90°. Superposition of two dipoles produces circulating current. Thus, in direction of the magnetic field, circularly polarized light is emitted.

For = +1, it is clockwise and

= –1, it is anti-clockwise circular

Common Data for Questions 50 and 51 :

The partition function for a gas of photons is given by

50. The specific heat of the photon gas varies with temperature as

(a)

(b)

(c)

(d)

Ans. (b)

Sol. The partition function of the photon gas is

The integral energy is

51. The pressure of the photon gas is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The free energy is

Statement for Linked Answer Questions 52 and 53:

Consider the propagation of electromagnetic waves in a linear, homogeneous and isotropic material medium with electric permittivity , and magnetic permittivity µ.

52. For a plane wave of angular frequency and propagation vector propagating in the medium Maxwell's equations reduce to

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

We know that Maxwell's equations,

53. If assume negative values in a certain frequency range, then the directions of the propagation vector and the Poynting vector in that frequency range are related as