SET-A

U/2015/18/II/A

SET-A

U/2015/18/II/A3

1. Which of the following function of complexvariable z = x iy is analytic—

(A) | z |

(B) Re z

(C) log z

(D) z–1

2. The value of the integral C

2sin(z) dz

z, is :

where countour C is a unit circle : |z – 2| = 1 is :

(A) 2i

(B) 4i

(C) i

(D) 0

3. The Legendre differential equation is solvedby series method. The equation given in—

22

2d y dy 3 3(1 x ) 2x 1 0

dx 2 2dx

The point when the solution will diverge if(1 – x2) = 0 and x is located at—

(A) 0 and 1

(B) 0 and –1

(C) –1 and 1

(D)32 and

52

4. Consider the differential equation Y' + p(x)Y'+ q(x) y | x | = 0 if xp(x) and x2q(x) have theTaylor series expansions

xp(x) = 4 + x + x2 + ........

x2p(x) = 2 + 3x + 5x2 + .......

Then the roots of the indicial equations are—

(A) –1, 0

(B) –1, –2

(C) –1, 1

(D) –1, 2

5. Given the recurrence relation for theLegendre polynomials—

(2n + 1)Pn(x) = (n + 1)Pn–1(x) + nPn–1(x)

Which of the integral has non-zero value—

(A)1

2n n 1

1x P (x)P (x)dx

(B)1

n n 21x P (x)P (x)dx

(C)1

2n

1x[P (x)] dx

(D)1

2n n 2

1x P (x)p (x) dx

SET-A

U/2015/18/II/A4

6. Consider a cylinder of height h and radius a,closed at both ends, centred at the origin. Let

ˆ ˆ ˆxi yj zk be the position vector and n̂ a isthe unit vector normal to the surface. The

surface integral S

ˆr .n ds

over the closed

surface of the cylinder is—

(A) 2a2 (a + h)

(B) 3a2h

(C) 2a2h

(D) Zero

7. A fourier series represented by f(x) = x sin xin the interval x , then the series

1 1 1 1 ......2 1.3 3.5 5.7

Will have the value.

(A) 2

(B) 4

(C) 6

(D) 8

8. If f(x) is analytic and single valued withinand on a closed contour c and if z0 is anypoint within c, then Couchy’s integralformula is—

(A) 00c

f (z)f (z ) dz(z z )

(B) 00c

1 f (z)f (z ) dz2 i (z z )

(C)0

0c

f (z )f (z ) dzz

(D)0

0c

f (z )1f (z ) dz2 i z

9. In a central force field, the trajectory of aparticle of mass m and angular momentumL in plane polar coordinates is given by

21 m (1 cos )r L , where is the

eccentricity of the particle’s motion, whichof the following for gives rise to aparabolic trajectory.

(A) = 0

(B) > 1

(C) = 1

(D) 0 < < 1

SET-A

U/2015/18/II/A5

10. A particle of mass M moves on a plane inthe field of a force given by F = ˆr̂ k r cos ,where k is constant and r̂ is the radial unitvector. The differential equation of the orbitof the moving particle will be—

(A) 2Mr Mr kr cos 0

(B) 2Mr Mr kr sin 0

(C) 2Mr Mr kr sin 0

(D) 2Mr Mr kr sin 0

11. A particle of mass m in attached to a thinuniform rod of length a and mass 4 m. Thedistance of the particle from the centre ofmass of the rod is a/4/ The moment of inertiaof the combination about an axis passingthrough—

Normal to the rod is—

(A) 291 ma48

(B) 251 ma48

(C) 227 ma48

(D) 264 ma48

12. For a particle moving in a central field—

(A) The kinetic energy is constant of motion

(B) The potential energy is velocity dependent

(C) The motion is confined in a plane

(D) The total energy is not conserved

13. Hamilton’s canonical equations of motionare—

(A) i ii i

H Hq and pp q

(B) i ii i

H Hq and pp q

(C) i ii i

H Hq and pp q

(D) i ii i

H Hq and pp q

14. A particle of mass m moves in a central force

field defined by F = – 4krr

, if E is the total

energy supplied to the particle, then the speedis given by—

(A) 2k 2E

mmr

(B) 2k 2E

mmr

(C) 2k 2E

mmr

(D) 2k 2E

mmr

SET-A

U/2015/18/II/A6

15. A particle constrained to move along the X-axis in a potential V = kx2 is subjected to anexternal time dependent force F

(t). Here kis a constant, x is the distance from the originand t is the time. At some time T, when theparticle has zero velocity at x = 0, the externalforce is removed. The particle will—

(A) Remain at rest

(B) Execute SHM

(C) Moving along positive x-direction

(D) Moving along negative x-direction

16. For a simple harmonic oscillator, theLagrangian is given by

2 21 1L q q ,2 2

if A(p, q) = p iq

2

and H(p, q) is the Hamiltonian of the system,the poison bracket {A(p, q), H(s, q)} is givenby—

(A) i A (p, q)

(B) A* (p, q)

(C) – iA* (p, q)

(D) –i A (p, q)

17. For the given transformation (1) Q = p andp = – q (2) p = q and Q = p; where p, q arecannomically conjugate variables, which oneof the following statements is true—

(A) Both (1) and (2) are cannonical

(B) Only (1) in canmonical

(C) Only (1) is cannonical

(D) Neither (1) nor (2)

18. A solid sphere of radius R carries a uniformvolume charge density P1 . The magnitude Athe electric field inside the sphere at adistance r from in centre is—

(A)0

R3t

(B)0

r3

(C)2

0

R3 r

(D)3

20

Rr

19. The electric | E |

and magnetic | B |

fieldamplitudes associated with an electromagneticradiation from a point source behave at adistance r from the source as—

(A) | E | consant, | B | = constant

(B) 21 1| E | , | B | r r

(C) 1 1| E | , | B | r r

(D) 3 31 1| E | , | B | r r

SET-A

U/2015/18/II/A7

20. Two point charges Q1 and Q2 are located atpoints A & B on a straight line as shown inthe figure—

The electric field will be zero at a point—(A) Between A and B(B) To the left of A(C) To the right of B(D) Perpendicular to AB

21. In an electromagnetic field, which one of thefollowing remains invariant under Lorentztransformations—

(A) 2 2E B

(B) E B

(C) 2B

(D) 2E

22. For good conductor, the average of poyntingvector is damped or attenuated as the waveprogresses with—

(A) n̂ˆ(n.r )2rmsE e

2 w

(B) n̂ˆ2 (n. r )2rmsE e

2 w

(C)ˆ2 (n. r )

rmsE e2 w

(D)ˆ2 (n.r )2 n̂

rmsE e2 w

Where is attenuation constant.

23. The electric flux passing through ahemispherical surface of radius R placed inuniform electric field E

with the axis parallel

to the field is—

(A) 2R E

(B) 22 R E

(C) 2 RE

(D) 32 R E

24. The potentials ( A

and ) at the position defined

by the vector r in an uniform electric field E

and uniform magnetic field B

will be—

(A)1E . r and A = (B r)2

(B)1E . r and A = (B r)r

(C) E. r and A = (B r )

(D) E . r and A = (B r)

SET-A

U/2015/18/II/A8

25. A particle has the wave function

(x, t) A exp(iwt) cos(kx)

which one of the following is correct—

(A) This is an eigen state of both energy and

momentum

(B) This is an eigen state A momentum and

not energy

(C) This is an eigen state of energy and not

momentum

(D) This is not an eigen state of energy or

momentum

26. A quantum harmonic oscillator is in the

energy eigen state n . A time independent

perturbation 1 (at a)2 acts on the particle,

where is a constant of suitable dimension,

and a & at are lowering and raising operators

respectively, then the first order energy shift

is given by—

(A) n

(B) 2n

(C) n2

(D) 2n2

27. In a Stern-Gerlach experiment, the magnetic

field is in +z direction. A particle comes out

of this experiment in z1 state. Which of

the following statement is true—

(A) A particle has a definite value of the y-

component of the spin angular momentum

(B) The particle has a definite value of the

square of the spin angular momentum

(C) The particle has a definite value of the

x-component of spin angular momentum

(D) The particle has definite value of x- and

y-component of spin angular momentum

28. Which of the following statements is/are true

for a linear harmonic oscillator—

(i) Energy levels are equally spaced

(ii) For lowest energy state the

probability is maximum at the centre

(i.e. origin)

(iii) Probability density distribution

shows maximum for nth energy state

(iv) Wave function corresponding to

lowest energy state is

1/ 22 2x / 2 2 mwe , where

h

(A) (i) and (ii) both are correct

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

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

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

SET-A

U/2015/18/II/A9

29. A particle in infinite potential wall of size‘a’ has the wave function as shown—

a(x, 0) Ax for 0 x 2

a(x, 0) A(a x) for x a2

The value of | A | is—

(A) 2 31a

(B) 2 32a

(C) 2 33a

(D) 32a

30. A particle is incident with a constant energyE on a one-dimensional potential barrier ofheight V (V > E > 0) as shown in the figure—

The wave functions in the region I, II and IIIare respectively—(A) Oscillatory, Exponentially-decaying,

Oscillatory(B) Oscillatory, Exponentially-decaying

Oscillatory, Oscilaltory with LowerAmplitude

(C) Oscillatory, Decaying, Oscillatory withsame amplitude

(D) Oscillatory, Exponentially-decayingOscillatory, Oscillatory with sameAmplitude

31. A particle of mass m moves in onedimensional simple harmonic potential V0 =

12 kx2 with angular frequency. W =

km

. AA

small perturbation term V(1) = 21 kx2 is

added to V0. What is the second orderperturbation to the ground state ?

(A)1 k w4 k

(B)1 k w4 k

(C)21 k w

16 k

(D)21 k w

16 k

32. A particle is incident with a constant energyE in a one dimensional potential barrier asshown—

The wave functions in region I and II arerespectively—

(A) Oscillator, Oscillatory

(B) Oscillatory, Decaying

(C) Decaying, Oscillatory

(D) Decaying, Decaying

SET-A

U/2015/18/II/A10

33. The expectation value of momentum of aparticle whose wave function is

2x ikx22ae(x) N

is given as—

(A) k

(B)ka

(C) 0

(D) 2h

a

34. If commutator [x, p] = i , the value of[x3, p] is—

(A) 22i x

(B) 22i x

(C) 23i x

(D) 23i x

35. Consider black body radiation in a cavitymaintained at 2000 k. If the volume of thecavity is adiabatically increased from 10 cm3

to 640 cm3, the temperature of the cavitychanges to—

(A) 800 K

(B) 700 K

(C) 600 K

(D) 500 K

36. A system of N non-interacting anddistinguishable particles of spin 1 is inthermodynamical equilibrium. The entropyof the system is—

(A) 2kBlnN

(B) 3kBlnN

(C) NkBln2

(D) NkBln3

37. The partition function of two Bose particles,each of which can occupy any of the twoenergy levels O and E is—

(A) 1 + e–2E/KT + e–E/KT

(B) 2 + e–2E/KT + e–E/KT

(C) 1 + e–2E/KT + 2e–E/KT

(D) e–2E/KT + e–E/KT

38. For an adiabatic change of an ideal gas, whichof the following remains constant—

(A) TVr – 1

(B) PVr – 1

(C) TPr – 1

(D) Tr – 1Pr

SET-A

U/2015/18/II/A11

39. Entropy remains constant in—

(A) Reversible isothermal process

(B) Reversible adiabatic process

(C) Reversible cyclic process

(D) Reversible isobaric process

40. Denoting temperature, entropy and internalenergy by T. S and U respectively and theirinitial and final values by subscripts i and frespectively, the Joule-Thomson expansionof an ideal gas can be expresses as—

(A) Uf Ui ; Tf = Ti ; Sf Si

(B) Uf Ui ; Tf = Ti ; Sf Si

(C) Uf Ui ; Tf Ti ; Sf Si

(D) Uf Ui ; Tf = Ti ; Sf Si

41. Which of the following relations betweeninternal energy U and the canomial partitionfunction z, is true—

(A) 2BU k T log z

T

(B) U log zT

(C) U kT log z

(D) U kT log zV

42. According to kinetic theory, the pressureexerted by a gas is given by the expression—(A) P = K.E. of all molecules in one mole of

the gas

(B) P = 13 K.E. of all molecules in one mole

of the gas

(C) P = 23 K.E. of all molecules in one mole

of the gas

(D) P = 34 K.E. of all molecules in one mole

of the gas

43. Let I1 and I2 represents mesh currents in theloop abcda and befcb respectively. The correctexpression describing Kirchhoff’s voltageloop law in one of the following loops is—

(A) 30 I1 – 15I2 = 10(B) –15I1 + 20I2 = –20(C) 30I1 – 15I2 = –10(D) –15I1 + 20I2 = 20

44. Which of the following statements isCORRECT for a common emitter amplifiercircuit—(A) Output is taken from the emitter(B) There is 180° phase shift between input

and output voltages(C) There is no phase shift between input

and output voltages(D) Both p-n junctions are forward biased

SET-A

U/2015/18/II/A12

45. The current flowing through the PNPtransistor of figure—

(A) 1 amp(B) 2 mA(C) 2.5 mAmp(D) None of these

46. In pn-junction with open ends—(i) There are no systematic motion of

charge carriers(ii) Holes and conduction electrons go

from p-side to n-side(iii) There is no net charge transfer

between the two sides(iv) There is a constant electric field near

the junction(A) (ii), (iii) and (iv) are correct(B) (i), (ii) and (iii) are correct(C) (i), (iii) and (iv) are correct(D) (i), (ii) and (iv) are correct

47. For an intrinsic semiconductor (Ge) at300 K, ni = 2.4 × 1019 m–3, mobility ofelectron µn = 0.39 m2–1s–1 and mobility ofholes µP = 0.19 m2–1s–1. The conductivityof sample is found to be—(A) 2.42 mho/m(B) 2.22 mho/m(C) 0.82 mho/m(D) 1.92 mho/m

48. For minimum offset voltage error due to biascurrent, the vlaue of R3 should be—

(A) R3 = R1 + R2

(B) R3 = 1 2

1 2

R RR R

(C) R3 = R1

(D) R3 = 2

1

RR

49. Standard error of correlation coefficient isgiven by the formula—

(A)n r1 n

(B) 21 rn

(C)21 r

n

(D)n r1 n

50. In the biased JFET, the shape of the channelM as shown in the given figure because—

(A) It is property of the terminal used(B) The drain end is more reversed biased

than source end(C) The drain end is more forward biased

than source end(D) The impurity profile varies with the

distance from the source

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U/2015/18/II/A

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U/2015/18/II/A

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U/2015/18/II/A

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U/2015/18/II/A