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U/2015/18/III/A

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U/2015/18/III/A3

1. The Fourier transform F(k) of a function f(x)is defined as

F(k) = dx f (x) exp (ikx)

then F(k) for

f(x) = exp(–x2) is—

2given : exp ( x )dx

(A) exp( k)

(B)2kexp

4

(C)2kexp

2 2

(D) 22 exp ( k )

2. If A and B are any two sets then which of thefollowing is correct—

(A) (A B) A B

(B) (A B) A B

(C) (A B) A B

(D) (A B) A B

3. In the group G = [E, A, A2], what is theelement conjugate to A2—

(A) E

(B) A

(C) A2

(D) None of these

4. How many independent components can beantisymmetric tensor of rank two in n-dimensional space—

(A) n2

(B) n(n + 1)

(C)n(n 1)

2

(D) 2n

5. The Green’s function for the operator L,

where 2

22

dL 0 x bdx

with

boundary conditions.

x b

Lt 1 d(t) 0 andt 0 dx

is given by—

(A) G(x, x') = x

x

cos h x e 0 x x

sin h x e x x b

(B) G(x, x') = x

x

sin h x e 0 x x

cos h x e x x b

(C) G(x, x') = x

x

sin h x e 0 x x

sin h x e x x b

(D) None of these

SET-A

U/2015/18/III/A4

6. One of the roots of the equation :

x3 – 3x2 + x – 3 = 0 is—

(A) – 1

(B) 1

(C) 3

(D) 3

7. What is the Laplace transform of square-wave function as shown in the followingfigure—

(A)A aSsin hS 2

(B)A aScos hS 2

(C)A aStanS 2

(D)A aStan hS 2

8. The order of error in the Simpson’s rule fornumerical integration with a step size h is—

(A) h

(B) h2

(C) h3

(D) h4

9. Equation of motion in Poisson bracket isgiven by—

(A) i i i i iq [q , H]; p [q , p ]

(B) i i i i iq [q , p ]; p [q , H]

(C) i i i iq [q , H]; p [q , H]

(D) i i i iq [q , H]; p [p , H]

10. If F is a vector that governs, the evolution

of a dynamical system, which of thefollowing is correct for a conservativedynamical system ?

(A) .F 0

(B) .F 0

(C) .F 0

(D) F 0

11. The dimensions of a generalized force Qi areML2T–2. The dimensions of the correspondinggeneralized coordinate qi and momentum piare respectively—

(A) 1 2 1 0 0 0M L T , M L T

(B) 0 0 0 1 2 1M L T , M L T

(C) 1 1 1 2M L T , M L T

(D) 2 1 1 1M L T , M L T

SET-A

U/2015/18/III/A5

12. In a Hamilton Jacabi equation H + st

= 0,

the Hamilton’s principle function s is—

(A) A function of generalized coordinates

and momenta pi = i

sq

(B) A function of generalized momenta and

coordinate qi = 2

2i

sp

(C) A function of generalized momenta andgeneralized coordinate

(D) A function of generalized coordinatesand S = H, the Hamiltonian

13. Which of the following are the generatingfunctions for the canonical transformation—

Q = (p2 + q2)1/2; P = –(p2 + q2)1/2 tan–1 qp

(i)2 2 11 q 1(p q ) tan qp

2 p 2

(ii)2 1 2 21 q 1Q sin q Q q

2 Q 2

(iii)2 1 2 2 1/ 21 P 1Q cos P(Q P )

2 Q 2

(iv)2 1 2 2 1/ 21 P 1Q cos p(Q P )

2 Q 2

(A) (i) and (iv) are correct

(B) (i), (ii) and (iii) are correct

(C) (i), (ii) and (iv) are correct

(D) (ii), (iii) and (iv) are correct

14. The value of the Poisson Bracket [a . r , b .p ] ,

where a and b

are constant vector isexpressed as—

(A) a b

(B) a b

(C) a b

(D) a .b

15. The Hamiltonian H(q, p, t) can be constructedusing the formula—

(A) i iH(q, p, t) p q L(q, q, t)

(B) i iH(q, p, t) p q L(q, q, t)

(C) H(q, p, t) T(q, q, t) V(q, q, t)

(D) H(q, p, t) 2T(q, q, t) L(q, q, t)

16. Find the correct form of cannonicaltransformation—

(A)y y

i ii i

F Fq ; Q

p p

(B)y y

i ii i

F Fq ; Q

p P

(C)y y

i ii i

F Fq ; Q

p P

(D)y

i ii i

F Fq ; Qp P

SET-A

U/2015/18/III/A6

17. In a system the generalized coordinates ‘q’and generalized momenta ‘p’ are transformedsuch that the new coordinates are—Q = p and P = –q, this transformation is—(A) Cannonical as Poisson bracket {Q, P}p, q = 1(B) Cannonical but Poisson bracket {Q, P}p, q = 0(C) Not a Cannonical transformation(D) Cannonical or non-cannonical depending

on the form of System’s Hamiltonian

18. Larmor formula for the power radiated bynon-relativistically accelerated chargedparticles is given by—

(A)2 2

30

1 2 e a4 3 c

(B)2 2

32 e a3 c

(C)2

20

1 2 e a4 3 c

(D)2 2

20

1 2 e a4 3 c

19. The Poynting vectors for velocity fields aregiven by—

(A) v 41S

R

(B) v 31S

R

(C) v 21S

R

(D) v1SR

20. The Lorentz transformations for E

and B

are—

(A) x x y y zE E ; E r(E vB )

(B) x x y y z2vB B ; B r B Ec

(C) x x y y z2vE E ; B r B Ec

(D) x x y y zB B ; E r(E v B )

where r = 2

2

1

1 vc

21. A long wire having a semi-circular loop ofradius r carries a current I as shown. Themagnetic induction at C due to entire wire—

(A) 02I

4 r

(B) 0I4r

(C) 02I

4r

(D) 0I4 r

SET-A

U/2015/18/III/A7

22. For a vector potential A , the divergence of

A is 0 QV.A

4 rV

where Q is a constant

of appropriate dimension. The corresponding

scalar potential (r, 0) that makes A &

Lorentz gauge invariant is—

(A)1 Q

4 v r

(B)1 Qt

4 v r

(C) 21 Q

4 v r

(D) 21 Qt

4 v r

23. This dispersion for a low density plasma in

w 2 = w02 + c2k2, where w0 is the plasma

frequency and c is speed of light in free space.

This relationship between group velocity (vg)

and phase velocity (vp) is—

(A) vp = vg

(B) vp = vg1/2

(C) vpvg = c2

(D) vg = vp1/2

24. The electric field E(r, t) at a point r at time tin a metal due to passage of E0ML waves canbe described by the equation

2L 2

2 21 E(r, t)v E( r , t) w E( r , t)c t

Where w is the characteristic associatedwith the metal and c is speed of light invacuum. The dispersion relationcorresponding to plane wave solution of the

form exp i(k . r wt) is given by—

(A) 2 2 2 2w c k w

(B) 2 2 2 2w c k w

(C) w ck w

(D) w ck w

25. The Lagrangian for a charged particle(charge +q) moving in an EM field withvelocity v is given by—

(A) 21 mv q( v.A)2

(B) 21 mv q( v.A)2

(C) 21 mv q( v.A)2

(D) 21 mv q( v.A)2

where v are scalar and vector potentials.

SET-A

U/2015/18/III/A8

26. Spin-orbit coupling splits 2p state in ahydrogen atom into two substates E apart,E is equal to—(A) 0(B) µBB(C) 2µBB(D) 3µBBwhere B is the induced magnetic field.

27. In Born approximation, the scatteringamplitude f(), when the potent ial inspherically symmetric, is—

(A)2

0

1 r sin rV(r)dr, with 2k sin2

(B)0

1 sin r V(r)dr, with 2k sin2

(C)0

1 r sin r V(r)dr, with 2k sin2

(D)

2

0

1 r sin r V(r)dr, with 2k sin2

28. The energy levels calculated using WKBapproximation of a spinless particle of massm moving in a one-dimensional box withwalls at x = 0 and x = L is—

(A)2 2

WKB 2n 2E n

2mL

(B)2 2

WKBn 2E n

2mL

(C) WKB 2n 2

1E n2mL

(D)2

WKBnE n

2mL

29. In the scattering of a particle from aspherically symmetric potential, theconserved quantities are—(A) E only(B) L2 only(C) E, L 2, Lz(D) L2, L2

30. For the rigid sphere of radius a, the scatteringcross section at low energy is given by—(A) 2a2

(B) a2

(C) 3a2

(D) 4a2

31. In relativistic quantum mechanics the energyoperator E satisfies—

(A) 2 2 2 2 2 4E ( c m c )

(B) 2 2 2 2 2 4E ( c 2m c )

(C) 2 2 2 2 2 4E ( c m c )

(D) 2 2 2 2 2 4E ( c m c )

where h2p

, h is Planck’s constant.

32. The Klein-Gordon equation describes—(A) Spinless particles(B) Spin half particle(C) Both spinless and spin half particles(D) Spin one particles

33. The Kein-Gordon equation gives the energyof a free particle as—(A) Positive(B) Negative(C) Both positive and negative(D) Imaginary

SET-A

U/2015/18/III/A9

34. The specific heat of the photon gas varieswith temperature as—

(A) 3VC T

(B) 2VC T

(C) 1VC T

(D) 2VC T

35. The free electron theory completely fails toexplain—(A) Heat capacity and paramagnetic

susceptibility(B) Ferromagnetic susceptibility(C) Heat capacity and ferromagnet ic

susceptibility(D) Diamagnetic susceptibility

36. Which of the following is present in all thesubstances—(A) Ferromagnetism(B) Paramagnetism(C) Diamagnetism(D) Anti-ferromagnetism

37. If the temperature is high and density is low,the number of bosons in ground level is—(A) Greater than other levels(B) Less than other levels(C) Same as other levels(D) None of the above

38. In the case of Bose-Einstein condensationnumber of particles in the lower energylevels—(A) Increases at low T and high P(B) Decreases at low T and high P(C) Increases at high T and low P(D) Decreases at high T and low P

39. Which of the following is true about rootmean square (rms) speed, average & speedand most probable speed—

(i) All are directly proportional tosquare root of temperature

(ii) All are inversely proportional tomolecular weight of gas

(iii) All are independent of molecularweight of gas

(iv) The ratio of rms speed : averagespeed : most probable speed is3 : 8 : 2

(A) (i) and (ii) are correct(B) (i) and (iii) are correct(C) (i) and (iv) are correct

(D) (i), (ii) and (iv) are correct

40. The T-S diagram of two cycles for an engineare shown (assume reversible cycles)—

The numerical values of T1, T2, S1 and S2 inthe two graphs is the same, which cycle hasgreater efficiency—

(A) A

(B) B

(C) Both have equal efficiency

(D) Insufficient information to decide

SET-A

U/2015/18/III/A10

41. The F-D occupation index is given by

F D 1i f

1f ( ) ( )kT 1

where f is called the ‘Fermi Energy’. The

value of F D 1f ( ) at the absolute zero of

temperature and i f is—

(A) 0

(B) 1

(C)

(D)12

42. The following circuit works as—

(A) RC-active low pass filter

(B) RC-active high pass filter

(C) RAMP generator

(D) RC-active medium pass filter

43. Which of the following advantages of the

pulse amplitude modulation (PAM)—

(i) Greater peak power compared to

amplitude modulation

(ii) Multiple signals may be transmitted

on a single carrier

(iii) Signal to noise ratio of the system is

impaired

(iv) Bandwidth remains constant even

after doubling the signal frequency

(A) (i) and (iv) are correct

(B) (ii) and (iii) are correct

(C) (ii) and (iv) are correct

(D) (i) and (iii) are correct

44. Which of the following is not a transducer—

(A) Accelerometer

(B) Loud speaker

(C) Rectifier

(D) Incandescent lamp

45. Which of the following statements is false

for inverting op-amplifier circuit—

(A) current into each terminal is zero

(B) Potential difference between input

terminals is zero

(C) A virtual short circuit exists at input

terminals

(D) Effectively an open circuit results

between input terminals

SET-A

U/2015/18/III/A11

46. If input auto-correlation function is

fwt0n

N1R e dw2 2

the out put power spectral density.

(A)0

2 2N

2(1 w )

(B)0

2 2N

2(1 w )

(C)20

2 2N

2(1 w )

(D)20

2 2N

2(1 w )

47. If the result of measurement is expressed as

X 3 , where X = mean value and standard deviation, it means that—

(A) Approximately 90% of the reading lie

between 3 limit

(B) 26 reading out of 1000 will lie outside

3 limit

(C) The odds for any reading to lie within

3 limit are 256 to 1

(D) All the above

48. A voltage signal v(t) has the followingFourier transform—

v(jw) = jwte for | w | 10 for | w | 1

the energy that will be dissipated in a 1 resistor fed from v(t) is—

(A)2 joules

(B)2d2e joules

2

(C)1

joules

(D)1

2 joules

49. For one population case, test of the variance

provides the equation of definite 2 as—

(A) 2 21p( )

2

(B) 2 2p( )2

(C) 2 2 21p( )

(D) 2 21p( )

2

50. The first line is the rotational spectrum ofCO is observed at x cm–1. The bond lengthof CO is proportional to—(A) x1/2

(B) x–1/2

(C) x(D) x–1

SET-A

U/2015/18/III/A12

51. The number of photons emitted per secondfrom a 1 Watt Ar-ion laser operating at488 nm is approximately—(A) 1.02 × 1030

(B) 2.46 × 1028

(C) 2.03 × 1018

(D) 2.46 × 1015

52. The NMR spectrum of ethanol (CH3CH2OH)comprises three bunches of spectral-lines inthe bunch corresponding to CH2 group is—(A) 1(B) 2(C) 3(D) 4

53. 1F3 – 1D2 is—(A) Singlet transition(B) Singlet-singlet transition and would give

rise to normal Zeeman triplet(C) Doublet-doublet transition and would

give rise to anomalous Zeeman effect(D) None of these

54. The total electronic angular momentum of aone-electron atom in the state 2D5/2 is—

(A)15h2

(B)35h4

(C)51h4

(D)45h2

55. The selection rule for transitions is—(A) jM 1, 2

(B) jM 0, 1

(C) jM 0, 1

(D) jM 1, 3

56. Three values of rotational energies ofmolecules are given below in different unitsP = 10 cm–1, Q = 10–23 J, R = 104MHz. choosethe correct arrangement in the increasingorder of energy—(A) P, Q, R(B) R, Q, P(C) R, P, Q(D) Q, R, P

57. The principal series of spectral lines oflithium is obtained by transitions between—(A) nS and 2p, n > 2(B) nD and 2p, n > 2(C) nP and 2S, n > 2(D) nF and 3D, n > 3

58. Infrared absorption for nitrogen molecules isabsent because—(A) Its dipole moment is zero(B) It has no vibrational level(C) it has no rotational level(D) Its polarizability is zero

59. Two important types of point defects are—(A) Edge and screw dislocations(B) Dislocations and vibrations(C) Vacancies and interstitials(D) Stacking faults

SET-A

U/2015/18/III/A

60. In molecular crystals the force between theatoms is—(A) Due to transfer of electrons(B) Due to sharing of electrons(C) Due to Vander Waals’ force arising out

of polarisation(D) Partly due to (A) & (B)

61. Inelastic scattering of photons in solids cangive rise to—(A) Only emission of phonons(B) Only absorption of phonons(C) Emission and absorption of phonons(D) No emission / absorption of phonons

62. In a tetragonal lattice a = b = 2.5 Å, c = 1.8Å. the lattice spacing between [111] planesis—(A) 6.32 Å(B) 1.85 Å(C) 1.26 Å(D) 0.62 Å

63. What will be the mobility of electrons incopper if there are 9 × 1028 valence electronsper (meter)3 and the conductivity A copper is6 ×107 mho per meter—(A) 6.6 × 103

(B) 5.5 × 10–3

(C) 4.16 × 10–3

(D) 3.2 × 10–3

64. The maximum energy that a free electron inmetal can have at absolute zero is—(A) Zero point energy(B) Fermi energy(C) Helmnoltz free energy(D) Gibb’s free energy

65. An electric field of 100 V/m is applied to asample of n-type semiconductor, whose Hallcoefficient is –0.0125 m3/C. If the electronmobility is 0.36 m2V–1s–1, the current densityin the sample is—(A) 2580 A/m2

(B) 2680 A/m2

(C) 2780 A/m2

(D) 2880 A/m2

66. A DC voltage of 5 µV is applied across theJosephson junction, the frequency of the ACcurrent produced is—(A) 2.41 × 108 Hz(B) 2.41 × 109 Hz(C) 2.41 × 1010 Hz(D) 2.41 × 1011 Hz

67. The Hall effect cannot explain—(A) The sign of current carrying charges(B) The number of charge carriers per unit

volume(C) Direct measurement of mobility(D) The effective mass of charges

68. Which of the following functions describesthe nature of interaction potential V(Y)between two quarks inside a nucleon ? (r isthe distance between the quarks and a & b arepositive constants of suitable dimensions—

(A) V(r) = a brr

(B) V(r) = a brr

(C) V(r) = a brr

(D) V(r) = a brr

13

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U/2015/18/III/A

69. Which of the following is true about nuclearforce—(A) n – n > n – p > p – p(B) n – n = n – p > p – p(C) n – n = n – p = p – p(D) n – n < n – p < p – p

70. Which is common in nuclear reactions—(A) r-rays(B) x-rays(C) Ultraviolet radiation(D) Infrared radiation

71. The unknown particle in the reaction givenbelow using conservation laws is—µ– + p 10n + ..........

(A) P

(B) P

(C) k0

(D) e–

72. According to liquid drop model theasymmetry energy term is negative and isproportional to—

(A)2(A 2Z)

Z

(B)2(A 2Z)

A

(C)2(A 2Z)

A

(D)2(A Z)

A

73. Magnetic moments of proton and neutronare—

(A) µp = 32 µN

µn = 1.91 µN

(B) µp = 2.7 µN

µn = 1.91 µN

(C) µp = 2.7 µN

µn = –1.91 µN

(D) µp = 13 µN

µn = 1.91 µN

where µN is nuclear magneton.

74. The experimentally measured spin g factorsof a proton and a neutron indicate that—

(A) Both proton and neutron are elementarypoint particles

(B) Both proton and neutron are notelementary point particles

(C) While neutron is an elementary pointparticle, proton is not

(D) While proton is an elementary pointparticle, neutron is not

75. The nucleus which is an isotope of 17Cl andalso an isobar of 18Ar has mass number Aand atomic number Z given by—

(A) A = 37 Z = 17

(B) A = 39 Z = 17

(C) A = 35 Z = 18

(D) Z = 37 Z = 19

14

SET-A

[SPACE FOR ROUGH WORK / ? ?? ??? ???? ???? ??????]

U/2015/18/III/A

SET-A

[SPACE FOR ROUGH WORK / ? ?? ??? ???? ???? ??????]

U/2015/18/III/A