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GATE2006

1. The trace of a 33 matrix is 2.Two of its eigenvalues are 1 and 2. The third eigenvalue is(a) -1 (b) 0 (c) 1 (d) 2

2. The value of C

ldA

. along a square loop of side L in a uniform field A

is

(a) 0 (b) 2LA (c) 4LA (d) L2 A

3. A particle of change q, mass m and linear momentum p

enters an electromagnetic field of vector

potentialA

and scalar potential .The Hamiltonian of the particle is

(a)

m

Aq

m

p

22

22

(b) m2

1qA

c

qp

2

(c) m2

1 2

Ac

qp

+ .p A

(d) Apq

m

p

2

2

4. A particle is moving in an inverse square force field If the total energy of the particle is positive,

then trajectory of the particle is

(a) circular (b) elliptical (c) parabolic (d) hyperbolic

5. In an electromagnetic field, which one of the following remains invariant under Lorentz

transformation?

(a) BE (b) 2E - 2c 2B (c) 2B (d) 2E

6. A space of radius R has uniform volume change density .the electric potential at a point r (r

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11. The target of an X-ray tube is subjected to an excitation voltage V. The wavelength of the emitted X-

ray is proportional to

(a)V

1 (b) V (c)

V

1 (d) V

12. The principal series of spectral lines of lithium 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

13. Which one of the following is NOT a correct statement about semiconductors ?

(a) The electrons and holes have different mobilities in semiconductor

(b) The an n-type semiconductor, the Fermi level lies closer to the conductor band edge

(c) Silicon is a direct band gap semiconductor

(d) Silicon has diamond structure

14. Which one of the following axes of rotational symmetry is NOT permissible in single crystals ?

(a) two-fold axis (b) three-fold axis (c) four-fold axis (d) five-fold axis

15. Weak nuclear forces act an

(a) both hadrons and leptons (b) hadrons only

(c) all particles (d) all charged particle

16. Which one of the following disintegration series of the heavy elements will give Bi209 as a stable

nucleus?

(a) Thorium series (b) Neptunium series

(c)Uranium series (d) Actinium series

17. The order of magnitude of the binding energy per nucleon in a nucleus is

(a) 10-5MeV (b) 10-3MeV (c) 0.1MeV (d) 10MeV

18. The interaction potential between two quarks separated by a distance rinside anucleon, can be described by (a band are positive constants )

(a) ae- r (b)r

a+br (c) -

r

a+br (d)

r

a

19. The high input impedance of field effect transistor (FET) amplifier is due to

(a) the pinch-off voltage (b) its very low gate current

(c) the source and drain being far apart

(d) the geometry of the FET

20. The circuit shown in the figure functions as

(a) an ORgate (b) an ANDgate

(c) an NORgate (d) a NANDgate

21. A linear transformation T, difined as T

3

2

1

x

x

x

=

32

21

xx

xx, transforms a vector x

from a three

dimensional real space to a two-dimensional real space. The transformation matrix T is

(a)

110

011 (b)

010

001 (c)

111

111 (d)

100

001

22. The value of S r

Sdr3

.

, where r

is the position vector and s is a closed surface

enclosing the origin ,is(a) 0 (b) (c) 4 (d) 8
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23. The value of dzze

C

z

4

2

1 , where C is a circle defined by z| =3,is

(a) 2

3

8 ei

(b) 1

3

8 ei

(c) ei

3

8 (d) 2

3

8ei

24. The thK Fourier component of xf = x is

(a)1 (b) 0 (c) 21

2 (d) 2

3

2

25. An atom with net magnetic mom

and net angular momentum LL

is kept in a uniform

magnetic induction kBB

0

.The magnetic moment

precesses about the z-axis. The equation of

motion of thex-component of x

uu

is

(a) 002

2

xx B

dt

d

(b) 002

2

xxx

dt

dB

dt

d

(c) 0202

2

2

xx B

dt

d

(d) 02 02

2

xx B

dt

d

26. The particle is moving in a spherically symmetric potential rV = 2r , where isa positive constant .in a stationary state ,the expectation value of the kinetic energy

of the particle is

(a) = (b) =2 (c) =3 (d) =4

27. A particle of mass 2 kg is moving such that at time t, its position ,in metre ,is given

by

jtitr 225)( . The angular moment of the particle at t=2s about the origin in kg ,12 sm is

(a) -40k

(b) -80 k

(c) 80 k

(d) 40k

28. A system of four particle is in x-y plane .Of these , two particles each of mass mare located at (-

1,1)and (1,-1).The remaining tow particles each of mass 2mare located at(1,1) and (-1,-1) .The xy-

component of the moment of inertia tensor of this system of particles is

(a) 10m (b) -10m (c) 2m (d) -2m

29. For the given transformations (i)Q = p ,P = -q, and (ii)Q = p, =qwhere pand qare canonically

conjugate variables ,which one of the following statement is true?

(a) both (i) and (ii) are canonical (b) only (i) is canonical(c) only (ii) is canonical (d) neither (i) nor (ii)is canonical

30. The mass mof a moving particle is3

2 0m ,where 0m is its rest mass. The linear

momentum of the particle is

(a) 2 0m c (b)3

2 0cm (c) 0m c (d)3

0cm

31. Three point charges q , qand -2qare located at (0,-a,a),(0,a,a)and (0,0,-a),respectively. The net dipole

moment of this charge distribution is

(a) 4qa k (b) 2qa k (c) -4qa i (d) -2qa j

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32. A long cylindrical conductor kept along z-axis carries a current density krJJ 0

, where 0J is a constant

and r is the radial distance from the axis of the cylinder. The magnetic induction B inside the conductor

at a distance dfrom the axis of the cylinder is

(a) 00J (b) -

2

00 dJ (c)

3

2

00 dJ (d) -

4

3

00 dJ

33. The vector potential in a region is given as jxiyzyxA 2),,(

.The associated magnetic induction

is B

is

(a) ji (b) k3 (c) ji 2 (d) kji

34. At the interface between two liner dielectrics (with dielectric constants 1 and 2 ), the electric field

lines bend as shown in the figure . Assume that there are no free charges at the interface .The ratio

21/ is

(a) 21 tan/tan (b) 21 cos/cos

(c) 21 sin/sin (d) 21 cot/cot

35. Which one of following sets of Maxwells equation for time-independent charge density and current

densityJ

is correct?

(a) E

. = 0/ ; 0. B

(b) 0/. E

; 0. B

t

BE

; t

EB

00 0 E

; JB

0

(c) 0. E

; 0. B

(d) 0/. E

; ||. 0 JB

0 E

; JB

0 0 E

;t

EB

00

36. A classical charged particle moving with frequencyin a circular orbit of radius a centred at the atthe origin in thex-yplane, emits electromagnetic radiation. At points (b,0,0)and (0,0,b),where b>>a,

the electromagnetic waves are

(a) circularly polarized and elliptically polarized ,respectively

(b) plane polarized and elliptically polarized ,respectively

(c) plane polarized and circularly polarized ,respectively(d) circularly polarized and plane polarized, respectively

37. A particle of mass mis represented by the wave function ikxAex ,where kiswave vector and A is a constant. The magnitude of the probability current density of the particle is

(a) 2A

m

k (b)

2A

m

k

2

(c)

2A

m

k 2

(d)

2A

m

k

2

2

38. A one dimensional harmonic oscillator is in the state xxxx 210 2314

1

Where )()(, 210 xandxx are the ground, first excited and second excited states respectively. The

probability of finding the oscillator in the ground state is

(a) 0 (b)14

3 (c)14

9 (d) 1

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39. The wavefunction of a particle in a one-dimensional potential at time t=0 is

xxtx 1021510, , where xandx 10 are the ground and the first excited state of

the particle with corresponding energies 10andEE .the wave function of the particle at a time t is

(a) xxetEEi

102

)10(

25

1

(b) xxe

tiE

10

0

25

1

(c) xxetiE

10

1

25

1

(d)

tiEtiE

exex1

1

0

025

1

40. The commutator [ yLx , ],where xL is the x-component of the angular momentum

operator andyis the y-component of the position operator is equal to

(a) 0 (b) xi (c) yi (d) zi 41. In hydrogenic states, the probability of finding an electron at r=0

(a) zero in state )(1 rs (b) non-zero in state s1 (r)

(c) zero in states2

(r) (d) non-zero in statep2 (r)

42. Each of the two isolated vessels, A and B of fixed volumes contains N, molecules of a perfect

monatomic gas at a pressureP.The temperatures of A, and B are 21andTT respectively. The two vessels

are brought into thermal contact. At equilibrium, the change in entropy is

a)

21

2

2

2

1

4ln

2

3

TT

TTNkB b) N Bk ln

1

2

T

T c)

21

2

21

4

)(ln

2

3

TT

TTNkB d) 2N Bk

43. The internal energy of n moles of a gas is givenvanRTE

23 ,where V is the volume of the gas at

temperature T and ais a positive constant. One mole of the gas in state 11,VT is allowed to expandadiabatically into vacuum to a final state 22 ,VT . The temperature 2T is

(a)

12

1

11

VVRaT (b)

12

1

11

3

2

VVRaT

(c)

12

1

11

3

2

VVRaT (d)

12

1

11

3

1

VVRaT

44. The mean internal energy of a one-dimensional classical harmonic oscillator in equilibrium with aheat bath of temperature T is

a) TKB2

1 b) TKB c) TKB

2

3 d) 3 TKB

45. A monatomic crystalline solid comprises ofNatoms ,out of which natoms are in

interstitial position. If the available interstitial sites are N , the number of possible microstates is

a)!!

)!(

Nn

nN b)

!!!

nNn

N

!!!

nNn

N

c)

!!!

nNn

N

d)

!!!

nNn

N

!!!

nNn

N

46. A system ofNlocalized, non-interacting spin ions of magnetic momenteach is

kept in an external magnetic field H. If the system is in equilibrium at temperature T, the Helmholtz

free energy of the system isa)

TK

HTNK

B

B

coshln b)

TK

HTNK

B

B

cosh2ln c)

TK

HTNK

B

B

cosh2 d)

TK

HTNK

B

B

sinh2ln

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47. The phase diagram of a free particle of mass mand kinetic energy E, moving in a one- dimensional

box with perfectly elastic walls atx=0andx=L, is given by

48. In the microwave spectrum of identical rigid diatomic molecules, the separation

between the spectral lines is recorded to be 0.7143 1cm .The moment of inertia of

the molecule in kg 2m is

a) 36103.2 b) 40103.2 c) 42108.7 d) 46108.7 49. Which one of the following electronic transitions in neon is NOT responsible for

LASER action in a helium-neon laser

a) 6s5p b) 5s4p c) 5s 3p d) 4s3p50. In the linear stark effect the application of an electric field

a) Completely lifts the degeneracy of n = 2 level of hydrogen atom and splits n = 2 level in to four

levels

b) partially lifts the degeneracy of n = 2 level of hydrogen atom and splits n =2 level into three levels

c) partially lifts the degeneracy of n = 2 level of hydrogen atom and splits n = 2 level into two levels

d) does not affect the n = 2 levels

51. In hyperfine interaction there is coupling between the electron angular momentum J

and nuclear

angular momentum I

,forming resultant angular momentum F

. The selection rules for the

corresponding quantum number F in hyperfine transitions area) 2F only b) 1F only c) 1,0F d) 2,1F

52. A vibrational-electronic spectrum of homonuclear binary molecules ,involving

electronic ground state and excited exhibits a continuum at 1cmv . If the total energy of thedissociated atoms in the excited state exceeds the total energy the dissociated atoms in the ground

state by 1cmEex , the dissociation energy of the molecule in the ground state is

a) 2/)( exEv b) 2/)( exEv c) )( exEv d) )( 22

exEv

53. The NMR spectrum of ethanol OHCHCH 23 comprises of three bunches of spectral lines. The

number of spectral lines in the bunch corresponding to )( 2CH group is

a) 1 b) 2 c) 3 d) 4

54. The energy KE

of electrons of wave vectorK

in a solid is given by

KE

= A 22 BKK , where A and B are constants. The effective mass of the electron at 0kk

is

a) A 20k b)A2

2

c)2

0

2

122 BkA

d)

2

0

2

Bk

55. Which one of the following statements is NOT correct about the Brillouin zone(BZ)

of a square lattices with constant a?

a) The first BZ is a square of saida

2 in yx kk plane

b) The area of first BZ and third BZ are the samec) The k-points are equidistant in

xk as well as in yk directions

d) The area of the second BZ is twice of the first BZ

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56. In a crystal of N primitive cells, each cell contains two monovalent atoms. The highest occupied energy

band of the crystal isa) one-fourth filled b) onethird filled c) half filled d) completely filled

57. If the number density of a free electron gas changes from 2628 1010 to electrons/ 3m , the value of plasma

frequency (in Hz) changes from 5.7 1510 to

a) 5.7 1310 b) 5.7 1410 c) 5.7 1610 d) 5.7 1710 58. Which one of the following statements about superconductors is NOT true?

a) A type I superconductor is complete diamagnetic

b) A type II superconductor exhibits Meissner effect upto the second critical

magnetic field2C

H

c) A type II superconductor exhibits zero resistance upto the second critical magnetic field

d) Both type I and type II superconductors exhibits sharp fall in resistance at the superconductingtransition temperature

59. Two dielectric materials A and B exhibits both ionic and orientational Polarizabilities. The variation

of their susceptibilities 1 r with temperature T is shown in the figure, where ris the relative dielectric constant .It can be inferred from the figure that

a) A is more polar and it has a smaller value of ionic polarizability then of B

b) A is more polar and it has a higher value of ionic polarizability then of B

c) B is more polar and it has a higher value of ionic polarizability than that of A

d) B is more polar and it has a smaller value of ionic polarozability than that of A

60. The experimentally measured spin g factors of proton and a neutron indicate that

a) both proton and neutron are elementary point particlesb) both proton and neutron are not elementary point particles

c) while proton is an elementary point particle, neutron is not

d) while proton is an elementary point particle, proton is not

61. By capturing an electron, 2954

25Mn transforms into 3054

25Cr releasing

a) a neutrino b) an antineutrino c) an -particles d) a positron62. Which one of the following nuclear processes is forbidden?

a) enp b) 0 ee c) KKnp d)

ee

63. To explain the observed magnetic moment of deuteron (0.8574 N ), its ground state wave function is

taken to be an admixture of S and D state. The expectation values of the z-component of the magnetic

in pure S and pure D states are 0.8797 N and 0.3101 ,N respectively. The contribution of the D stateto the mixed ground state is approximately

a) 40% b) 4% c) 0.4% d) 0.04%

64. A sinusoidal input voltage in of frequency is fed to the circuit shown in the

figure, where 21 CC . If m is the peak value of the input voltage, then output

voltage ou t isa) 2 m b) 2 0 tsin

c) 2 m d)2

m tsin

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65. The low-pass active filter shown in the figure has a cut-off frequency of 2kHz and

a pass band gain of 1.5. The values of the resistors area) 3.1;10 21 RkR

b) 3.1;30 21 RkR

c) 7.1;10 21 RkR d) 7.1;30 21 RkR

66. In order to obtain a solution of the differential equation 02 12

2

dt

d

dt

d

involving voltages ,)( 1vandtv an operational amplifier (Op-Amp) circuit would require at least

a) two Op-Amp integrators and one Op-Amp adder

b) two Op-Amp differentiators and one Op-Amp adder

c) one Op-Amp integrators and one Op-Amp adder

d) one Op-Amp integrator, one Op-Amp differentiator and one Op-Amp adder

67. In the given digital logic circuit, A and B form the input. The output Y is

a) AY b) BAY

c) BAY d) BY 68. The largest analog output voltage from a 6-bit digital to analog converter (DAC)

which produces 1.0V output for a digital input of 010100, is

a) 1.6V b) 2.9V c) 3.15V d) 5.0V

69. A ripple counter designed which JK flip flops provided with CLEAR(CL)input is shown in the figure.

In order that this circuit function as a MOD-12 counter, the NAND gate input( 21andXX )should be

a) A and C b) A and D

c) B and D d) C and D

70. The tank circuit of a Hartley oscillator is shown in figure. If M is the mutual inductance between the

inductors, the oscillation frequency is

a)CMLL )2(2

1

21 b)

CMLL )2(2

1

21

c)CMLL )(2

1

21 d)

CMLL )(2

1

21

Common Data for Questions 71 to 73

An unperturbed two-level system has energy eigen values 1E and ,2E and

eigen functions

0

1 and

1

0. When perturbed, its Hamiltonian is represented

by

2

1

EA

AE

71. The first-order correction to 1E is

a) 4A b)2A c)A d) 0

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72. The second-order correction to 1E is

a) 0 b) A c)12

2

EE

A

d)

21

2

EE

A

73. The first-order correction to eigenfunction

0

1is

a)

21

0

EEA

b)

1

0 c)

021 EE

A d)

1

1

Common data for questions 74, 75:

One of the eigenvalues of the matrix

100

023

032 is 5

74. The other two eigenvalues are

a) 0 and 0 b) 1 and 1 c) 1 and-1 d) -1 and -1

75. The normalized eigenvector corresponding to the eigenvalue 5 is

a)

1

1

0

2

1 b)

0

1

1

2

1 c)

1

0

1

2

1 d)

0

1

1

2

1

Linked Answer 76 to 77

The power diffraction pattern of a body centred cubic crystal is recorded by using CuKX-rays of

wavelength 1.54Ao.

76. If the (002) planes diffract at 60 , then lattice parameter is

a) 2.67 A b) 3.08 A c) 3.56 A d) 5.34 A

77. Assuming the atomic mass of the constituent atoms to be 50.94amu, then density of crystal in units of

kg 3m is

a) 3.75 310 b) 4.45 310 c) 5.79 310 d) 8.89 310 Statement for Linked Answer Questions 78 & 79:

A particle of mass m is constrained to move in a vertical plane along a trajectory

given by x=A cos , y=A sin , where A is a constant

78. The Lagrangian of the particle is

a) mgmA 222

1 A cos b) mgmA 22

2

1 A sin

c) 22

2

1mA d) mgmA 22

2

1 A cos

79. The equation of motion of the particle is

a) 0cos A

g b) 0sin A

g c) =0 d) 0sin A

g

Statement for Linked Answer Questions 80 & 81:

A dielectric sphere of radius R carries polarization ,2rkrP

where r is the distance from the centre

and k is a constant. In the spherical polar coordinate system, ,r and are the unit vectors.

80. The bound volume charge density inside the sphere at a distance r from the centre isa) -4kR b) -4kr c) -4k 2r d) -4k 3r

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81. The electric field inside the sphere at a distance d from the centre is

a) rkd 0

2

b) rkR

0

2

c)

0

2kd d)

0

2kR

Linked 82 and 83

Consider Fermi theory of -decay.

82. The number of final states of electrons corresponding to momenta betweenp and p + dpis

a) independent of p b) proportional to pdp

c) proportional to dpp2 d) proportional to dpp3

83. The number of emitted electrons with momentum p and energy E, in the allowed approximation, is

proportional to ( 0E is the total given up by the nucleus)

a) EE 0 b) p EE

0 c)

2

0

2

EEp

d)

2

0 EEp

Linked 84-85

Consider a radiation cavity of volume V at temperature T.

84. The density of states at energy E of the quantized (photons) is

a) 223

8E

ch

V b) 2

3

33

8E

ch

V c) E

ch

V23

8 d) 2

1

23

8E

ch

V

85. The average number of photons in equilibrium inside the cavity is proportional to

a) T b) 2T c) 3T d) 4T

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