ugc point academy · 2019-04-13 · coordinate and , a and r are positive constants. if the...

27-G 2ND FLOOR, JIA SARAI, NEAR IIT, NEW DELHI- 110016 Tel: 011-26521410, 26855515 Mobs: 09654680505, 07503646974

E-mail: [email protected] Website: www.ugcpoint.in

UGC POINT ACADEMY LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM

CSIR-UGC-NET/JRF DEC-2014 PHYSICAL SCIENCES DEC 2014

PART–B

21. Let denote the position vector of any point in three-dimensional space, and , then

(1) and (2) and

(3) and (4) and

22. The column vector is a simultaneous eigenvector of and if

(1) or (2) or

(3) or (4) or

23. The principal value of the integral is

(1) (2) (3) (4)

24. The Laurent series expansion of the function about z=0 is given by:

(1) for all (CSIE-NET-DEC-2014)

(2) only if

(3) for all

(4) if

25. Two independent random variables and , which can take the integer values 0, 1, 2, …….. , follow

the Poisson distribution, with distinct mean values and respectively then

(1) The probability distribution of the random variable is a binomial distribution

(2) The probability distribution of the random variable is also a Poisson distribution

(3) The variance of the random variable is equal to

(4) The mean value of the random variable is equal to 0

r r r

. 0r /r r r . 0r 2 0r

. 0r 2 2/r r r . 3r 0r

a

b

a

0 0 1

0 1 0

1 0 0

A

0 1 1

1 0 1

1 1 0

B

0b 0a b a 2b a

2b a b a / 2b a / 2b a

3

sin 2xdx

x

2 2

2 1/ zf z e e

n!

n

n

z z

0

1 1

!

nn n

znz

0 | | 1z

0

1 1

!

nn n

znz

0 | |z

0

!

n

n

zonly

n | | 1z

m n

, v

l m n

r m n

l m n v

r m n



26. The equation of motion of a system described by time-dependent

Lagrangian is

(1) (2)

(3) (4)

27. A particle of mass m is moving in the potential where a, b are positive constant.

The frequency of small oscillations about a point of stable equilibrium is

(1) (2) (3) (4)

28. The radius of Earth is approximately 6400 km. The height h at which the acceleration due to Earth’s

gravity differs from g at the Earth’s surface by approximately 1% is:

(1) 64 km (2) 48 km (3) 32 km (4) 16 km

29. According in the theory of relativity, the speed v of a free particle of mass m and total energy E is:

(1) (2) (3) (4)

30. A charged particle moves in a helical path under the influence of a constant magnetic field. The initial

velocity is such that the component along the magnetic field is twice the component in the plane normal

to the magnetic field.

The ratio of the pitch to the radius R of the helical path is

(1) (2) (3) (4)

31. A parallel beam of light of wavelength is incident normally on a thin polymer film with air on both

sides. If the film has a refractive index n > 1. Then second order bright can be observed in reflection

when the thickness of the film is

(1) (2) (3) (4)

32. A solid sphere of radius R has a charge density, given by , where r is the radial

coordinate and , a and R are positive constants. If the magnetic of the electric field at is 1.25

times that at , then the value of is

(1) 2 (2) 1 (3) 1/2 (4) 1/4

21

2

tL e mx V x

0dV

mx mxdx

0dV

mx mxdx

0dV

mx mxdx

0dV

mxdx

2 41 1

2 4V x ax bx

/a m 2 /a m 3 /a m 6 /a m

2

1mc

v cE

22

1E mc

vm E

22

1mc

v cE

2

1mc

v cE

/ R

2

4 2

/ 4n / 2n 3 / 4n / n

0 1ar

rR

0 / 2r R

r R a



33. The electrostatic lines of force due to a system of four point charges is sketched below

at a large distance r, the leading asymptotic behavior of the electrostatic potential is proportional to

(1) (2) (3) (4)

34. Suppose Hamiltonian of a conservative system in classical mechanics is where is a constant

and and p are the position and momentum respectively. The corresponding Hamiltonian in quantum

mechanics, in the coordinate representation, is

(1) (2) (3) (4)

35. Let and denote the normalized eigenstates of a particle with energy eigenvalues and

respectively, with . At time the particle is prepared in a state . The

shortest time T at which will be orthogonal to is

(1) (2) (3) (4)

36. Consider the normalized wave function where is a simultaneous normalized

eigen function of the angular momentum operators and , with eigenvalues and

respectively. If is an eigenfunction of the operator with eigenvalue , then

(1) (2)

(3) (4)

37. Let and denote, respectively, the coordinate and momentum operators satisfying the canonical

commutation relation in natural unit . Then the commutator is:

(1) (2) (3) (4)

38. The pressure P of a fluid is related to its number density by the equation state where

and are constants. If the initial volume of the fluid is the work done on the system when it is

compressed so as to increases

(1) (2) (3) (4)

r 1r 2r 3r

,H xp

x

1

2i x

x

1

2i x

x

i x

x

2

ix

x

1 2 1E 2E

2 1E E 0t 1 2

10

2t

t T 0t

2 1

2

E E

2 1E E

2 12 E E

2 14 E E

1 11 2 10 3 1 1a a a m

2L zL 21l l m

xL

1 3 2

1 1,

2 2a a a 1 3 2

1 1,

2 2a a a

1 3 2

1 1,

2 2a a a 1 2 3

1

3a a a

x p

,x p i 1 , px pe

1 pi p e 21 pi p e 1 pi epipe

2P a b a

b 0 ,V

0 0a V 0 0 0a b V 00 0

73

2 3

baV

0 0 0ln2a b V



39. The Hamiltonian of a classical particle moving in one dimension is where is a positive

constant and and q are its momentum and position respectively. Given that its total energy the

available volume of phase space depends on as

(1) (2) (3) (4) is independent of

40. An ideal Bose gas is confined inside a container that is connected to a particle reservoir. Each particle

can occupy is discrete set of single particle quantum states. If the probability that a particle quantum

state is unoccupied is 0.1, then the average number of bosons in that state is

(1) 8 (2) 9 (3) 10 (4) 11

PART-C

41. In low density oxygen gas at low temperature, only the translational and rotational modes of the

molecules are excited. The specific heat per molecule of the gas is

(1) (2) (3) (4)

42. Consider the amplifier circuit comprising of the two op-amps and as shown in the figures

If the input ac signal source has an impedance of , which of the following statements is true?

(1) is required in the circuit because the source impedance is much greater than .

(2) is required in the circuit because the source impedance is much less than R.

(3) can be eliminated from the circuit without affecting the overall gain,

(4) is required in the circuit if the output has to follow the phase of the input signal

43. The I-V characteristics of the diode in the circuit below is given by

where V is measured in volts and I is measured in amperes

The current I in the circuit is:

(1) 10.0 mA (2) 9.3 mA (3) 6.2 mA (4) 6.7 Ma

24

2

pH q

m

p 0E E

0E

3/40E 0E 0E

0E

1

2Bk Bk

3

2Bk

5

2Bk

1A 2A

50k

1A r

1A

1A

1A

0.7 / 500 0.7

0 0.7

V for VI

for V



44. A junction is made between a metal of work function and a doped semiconductor of work function

with . If the electric field at the interface has to be increased by a factor of 3, then the

dopant concentration in the semiconductor would have to be.

(1) Increased by a factor of 9 (2) decreased by a factor of 3

(3) Increased by a factor of 3 (4) decreased by a factor of

45. In a measurement of the viscous drag force experienced by spherical particles in a liquid, the force is

found to be proportional to where V is the measured volume of each particle. If V is measured to be

30 mm3, with an uncertainty of 2.7 mm

3, the resulting relative percentage uncertainty in the measured

force is:

(1) 2.08 (2) 0.09 (3) 6 (4) 3

46. Let , where are the Pauli matrices. If and are two arbitrary constant vectors

in three dimensions the commutator is equal to (in the following I is the identity matrix)

(1) (2)

(3) (4)

47. Consider the function of a complex variable . The singularities

of are as follows:

(1) Branch points at and ; and a pole at only for

(2) Branch points at and ; and a pole at for all other than

(3) Branch points at and ; and a pole at for all

(4) Branch points at and

48. The function satisfies the differential equation

(1)

(2)

(3)

(4)

49. Let and be complex number which of the following sets of matrices forms a group under matrix

multiplication?

(1) (2) where

(3) where is real (4) where

3PW

5 ,W M SW W

3

1/3V

1 2 3, , 1 2 3, , a b

. ,b.a

1 2 3.a b 2 .i a b

.a b I a b I

1

1f z n zz

,iz re r

f z

1z z 0z 0 2

1z z 0z 0 2

1z z 0z

0, 1z z z

2 1

0

1

! 1 ! 2

n n

n

xf x

n n

2

2 2

21 0

d f dfx x x f

dxdx

2

2 2

22 1 0

d f dfx x x f

dxdx

2

2 2

21 0

d f dfx x x f

dxdx

2

2 2

21 0

d f dfx x x f

dxdx

0 0

1,

1

1

*,

*

* ,* *

2 2| | | | 1



50. The expression (where is the Levi-Civita symbol, are the position,

momentum and angular momentum respectively, and represents the Poisson Bracket of A and B)

simplifies to

(1) 0 (2) 6 (3) (4)

51. A mechanical system is described by the Hamiltonian . As a result of the

canonical transformation generated by the Hamiltonian in the new coordinate Q and

momentum P becomes

(1) (2)

(3) (4)

52. The probe Mangalyaan was sent recently to explore the planet Mars. The inter-planetary part of the

trajectory is approximately a half-ellipse with the Earth(at the time of launch), Sun and Mars (at the time

the probe reaches the destination) forming the major axis. Assuming that the orbits of Earth and Mars

are approximately circular with radii RE and RM, respectively, the velocity (with respect to the Sun) of

the probe during its voyage when it is at a distance from the Sun, neglecting the effect

of Earth and Mars is

(1) (2)

(3) (4)

53. A plane electromagnetic wave incident normally on the surface of a material is partially reflected.

Measurements on the standing wave in the region in front of the interface show that the ratio of the

electric field amplitude at the maxima and the minima is 5. The ratio of the reflected intensity to the

incident intensity is

(1) 4/9 (2) 2/3 (3) 2/5 (4) 1/5

3

, , 1

, ,ijk i j k

i j k

x p L

ijk , ,x p L

,A B

.x p L x p

2

2 21,

2 2

pH q p m q

m m

, ,Q

F q Qq

22 2 21

2 2

mQ P Q

m

22 2 21

2 2

mQ P P

m

22 21

2 2

mP Q

m

22 4 21

2 2

mQ P P

m

E Mr R r R

2

E M

E M

R RGM

r R R r

2

E M

E M

R R rGM

r R R

2 E

M

RGM

rR

2GM

r



54. The scalar and vector potential and are determined up to a gauge transformation

and where is an arbitrary continuous and differentiable function of and

. If we further impose the Lorenz gauge condition then a possible choice for the gauge

function is (where are nonzero constants with )

(1) (2)

(3) (4)

55. A non-relativistic particle of mass m and charge e, moving with a velocity and acceleration , emits

radiation of intensity I. What is the intensity of the radiation emitted by a particle of mass m/2 charge 2e,

velocity and acceleration ?

(1) 16 I (2) 8I (3) 4I (4) 2I

56. The ground state energy of the attractive delta function potential where b > 0, is

calculated with the variational trial function

(1) (2) (3) (4)

57. Let (where and are constants with )be a linear combination of the wave-

function of the ground and first excited states of the one-dimensional harmonic oscillator. For what

value of is the expectation value is maximum?

(1) (2)

(3) (4)

58. Consider a particle of mass m in the potential . The energy eigen-values

, in the WKB approximation are

(1) (2)

(3) (4)

,x t ,A x t

't

'A A A x

t1

. 0Ac t

,x t ,k c k

cos cosh .t k x sinh t cosk.x

cosh cos .t k x cosh cosh .t k x

v a

/ 2v 2a

,V x b x

cos ,

2

0,

xA for a x a

x a

otherwise

2

2 2

mb

2

2 2

2mb

2

2 22

mb

2

2 24

mb

0 10 1c c 0c 1c2 20 1 1c c

0c x

0

1,

2x c

m 0

1,

2 2x c

m

0

1,

2 2x c

m 0

1,

2x c

m

| |, 0V x a x a

0,1,2,3....nE n

1/33 1

24 2

an

m

2/33 1

24 2

an

m

3 1

24 2

an

m

4/33 1

24 2

an

m



59. The Hamiltonian for a three-state quantum system is given by the matrix . What

perturbed by where , the resulting shift in the energy eigenvalue is

(1) (2) (3) (4)

60. When a gas expand adiabatically from volume to by a quasi-static process, it cools from

temperature to . If now the same process is carried out adiabatically and irreversibly, and is the

temperature of the gas when it has equilibrated, then

(1) (2) (3) (4)

61. A random Walker takes a step of unit length in the positive direction with probability 2/3 and a step of

unit length in the negative direction with probability 1/3. The mean displacement of the walker after n

step is

(1) (2) (3) (4)

62. A collection, N of non-interacting spins is kept in an external magnetic field B at

a temperature T. The Hamiltonian of the system is . What should be the minimum value of

for which the mean value

(1) (2) (3) (4)

63. A large MOS transistor consists of N individual transistors connected in parallel. If the only form of

noise in each transistor is 1/f noise, then the equivalent voltage noise spectral density for the MOS

transistor is

(1) 1/N times that of a single transistor (2) 1/N2 times that of a single transistor

(3) N times that of a single transistor (4) N2 times that of a single transistor

64. Consider a Low Pass(LP) and a High Pass(HP) filter with cut off frequencies and respectively,

connected in series or in parallel configuration as shown in the figures A and B below

Which of the following statements is correct?

0H 0

1 0 0

0 2 0

0 0 2

H

0 1 0

' 1 0 1

0 1 0

H

1 0 2E

, 2 , 2 2

1V 2V

1T 2T 2 'T

'2 2T T '

2 2T T ' 2 12 2

2

V VT T

V

' 2 12

2

T VT

V

/ 3n / 8n 2 / 3n 0

1,2......, , 1i iS i N S

i iH B S

B

B

k T

1?

3iS

1ln 2

2N 2ln 2

1ln 2

2ln2N

LPf HPf



(1) For A acts as a Band Pass filter and B acts as a Band Reject filter

(2) For A stops the signal from passing through and B passes the signal without filtering

(3) For A acts as a Band Pass filter and B passes the signal without filtering

(4) For A passes the signal without filtering and B acts as a Band Reject filter

65. When laser light of wavelength falls on a metal scale with 1 mm engravings at a grazing angle of

incidence, it is diffracted to form a vertical chain of diffraction spots on a careen kept perpendicular to

the scale. If the wavelength of the laser is increased by 200 nm the angle of the first-order diffraction

spot changes from 5o to

(1) (2) (3) (4)

66. The power density of sunlight incident on a solar cell is 100mw/cm2. Its short circuit current density is

30mA/cm2 and the open circuit voltage is 0.7V. if the fill factor of the solar cell decreases from 0.8 to

0.5 then the percentage efficiency will decreases from

(1) 42.0 to 26.2 (2) 24.0 to 16.8 (3) 21.0 to 10.5 (4) 16.8 to 10.5

67. An atomic transition in a magnetic field 1 Tesla shows Zeeman splitting. Given that the Bohr

magneton , and the wavelength corresponding to the transition is 250 nm, the

separation in the Zeeman spectral lines is approximately

(1) 0.01 nm (2) 0.1 nm (3) 1.0 nm (4) 10 nm

68. If the leading anharmonic correction to the energy of the nth

vibrational level of a diatomic molecule is

with , the total number of energy levels possible is approximately

(1) 500 (2) 1000 (3) 250 (4) 750

69. The effective spin-spin interaction between the electron spin and the proton spin in the ground

state of the Hydrogen atom is given by . As a result of this interaction, the energy levels split

by an amount

(1) (2) (3) (4)

70. Consider the crystal structure of sodium chloride which is modelled as a set of touching spheres. Each

sodium atom has a radius r1 and each chlorine atom has a radius r2. The centres of the spheres form a

simple cubic lattice. The packing fraction of this system is

(1) (2)

(3) (4)

HP LPf f

HP LPf f

HP LPf f

HP LPf f

6.60o 5.14o 5.108o 5.21o

1 1P S

249.27 10 /B J T

21

2ex n

0.001ex

eS pS

' .e pH aS S

21

2a 22a 2a

23

2a

3 2

1 2

1 2 1 2

r r

r r r r

3 31 2

3

1 2

2

3

r r

r r

3 31 2

3

1 2

r r

r r

3 31 2

3

1 22

r r

r r



71. Consider two crystalline solids, one of which has a simple cubic structure and the other has a tetragonal

structure. The effective spring constant between atoms in the c-direction is half the effective spring

constant between atoms in the a and b directions. At low temperatures the behaviour of the lattice

contribution to the specific heat will depend as a function of temperature T as

(1) for the tetragonal solid, but as for the simple cubic solid

(2) T for the tetragonal solid and as for the simple cubic solid

(3) T for both solids

(4) for both solid

72. A superconducting ring carries a steady current in the presence of a magnetic field normal to the plane

of the ring, identify the incorrect statement.

(1) The flux passing through the superconductor is quantized in units of he/e

(2) The current and the magnetic field in the superconductor are time independent

(3) The current density and are related by the equation where A is a constant

(4) The superconductor shows an energy gap which is proportional to the transition temperature of the

superconductor

73. Consider the four processes

(i) (ii)

(iii) (iv)

Which of the above is/are forbidden for free particles

(1) Only (ii) (2) (ii) and (iv) (3) (i) and (iv) (4) (i) and (ii)

74. In deep inelastic scattering electrons are scattered off protons to determine if a proton has any internal

structure. The energy of the electron for this must be at least

(1) (2) (3) (4)

75. If the binding energy B of a nucleus (mass number A and charge Z) is given by

where

(1) 68 (2) 72 (3) 84 (4) 92

2T 3T3T

3T

B

J B 2 0,J A B

ep n e v 0ep e v

ee v 0

91.25 10 eV 121.25 10 eV 61.25 10 eV 81.25 10 eV

2 2

2/3

1/3

2C

V S sym

Z A a ZB a A a A a

A A

16 , 16v sa MeV a MeV

ugc point academy · 2019-04-13 · coordinate and , a and r are positive constants. if the...

Documents