Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE …...fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES fiziks c/o Anand Institute of mathematics, 28-B/6 Jia

fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM, GRE in PHYSICAL SCIENCES

fiziks c/o Anand Institute of mathematics, 28-B/6 Jia Sarai

Near IIT, Hauz Khas, New Delhi, PIN- 110016 (INDIA) Phone: 011-32718565, +91-9871145498

Website: http://www.physicsbyfiziks.com 1 Email: [email protected]

GS-2010 (Physics)

TATA INSTITUTE OF FUNDAMENTAL RESEARCH

Written Test in PHYSICS - December 19, 2009

Duration: Three hours (3 hours)

Name: ________________________________ Ref. Code: ____________

Please read all instructions carefully before you attempt the questions.

1. Please fill-in details about name, reference code etc. on the question paper and answer

sheet. The Answer Sheet is machine-readable. Read the instructions given on the reverse

of the answer sheet before you start filling it up. Use only HB pencils to fill-in the answer

sheet.

2. Indicate your ANSWER ON THE ANSWER SHEET by blackening the appropriate

circle for each question. Do not mark more than one circle for any question: this will be

treated as a wrong answer.

3. This test comes in two sections, Section A and Section B, both of which contain

multiple choice-type questions. Only ONE of the options given at the end of each

question is correct. Section A contains 20 questions, each with 4 options, and Section B

contains 8 questions, each with 6 options. The maximum marks are 60 for Section A plus

40 for Section B, totaling to 100. Marking shall be as follows:

(i) If the answer is correct: +3 marks in Section A; +5 marks in Section B

(ii) If the answer is incorrect: -1 mark in both Section A & B

(iii) If the answer is not attempted: 0 marks in both Section A & B

(iv) If more than one box is marked: 0 marks in both Section A & B

Note that negative marking as indicated above will be implemented.





4. As a rough guideline, the time spent on questions in Section A should be about 5

minutes each; questions in Section B should take about 10 minutes each. Obviously,

some questions may take a little less time while others may require a little more.

5. We advise you to first mark the correct answers on the QUESTION PAPER and then

to TRANSFER these to the ANSWER SHEET only when you are sure of your choice.

6. Rough work may be done on blank pages of the question paper. If needed, you may

ask for extra rough sheets from an Invigilator.

7. Use of calculators is permitted. Calculator which plots graphs is NOT allowed.

Multiple-use devices such as cell phones, smart phones etc., CANNOT be used for this

purpose.

8. Do NOT ask for clarifications from the invigilators regarding the questions. They have

been instructed not to respond to any such inquiries from candidates. In case a

Correction/clarification is deemed necessary; the invigilator(s) will announce it publicly.





GS-2010 (Physics)

A SECTION: 20 x 3= 60 Marks

Q.1 The matrix

110111011

can be related by a similarity transformation to the matrix

(a)

011111111

(b)

210

111012

(c)

110111011

(d)

110111011

Q.2 A car tyre is slowly pumped up to a pressure of 2 atmospheres in an environment at

15o C. At this point, it bursts. Assuming the sudden expansion of the air (a mixture of O2

and N2) that was inside the tyre to be adiabatic, its temperature after the burst is

(a) – 55o C (b) – 37o C (c) – 26o C (d) + 9o C

Q.3 A small meteor approaches the Earth. When it is at a large distance, it has velocity v∞ and

impact parameter b. If Re is the radius of the Earth and v0 is the escape velocity, the

condition for the meteor to strike the Earth is

(a) 20 /1 vvRb e (b) 20 /1 vvRb e

(c) 20 /1 vvRb e (d) vvRb e /0

Q.4 Consider a very, very thin wire of uniformly circular cross section. The diameter of the

wire is of the order of microns. The correct equipment required to measure the precise

value of resistivity of this wire is

(a) ammeter, voltmeter, scale, slide calipers

(b) ammeter, magnet, screw gauge, thermometer

(c) voltmeter, magnet, screw gauge, scale

(d) ammeter, voltmeter, scale, monochromatic laser source





Q.5 A function f(x) is defined in the range – 1 ≤ x ≤ 1 by

f(x) = 0for 10for 1

xxxx

The first terms in the Fourier series approximating this function are

(a) .......3cos9

4cos421

22 xx

(b) .......3sin9

4sin421

22 xx

(c) .......3cos9

4cos422 xx

(d) .......3cos9

4cos421

22 xx

Q.6 A lead container contains 1 gm of a Co6027 radioactive source. It is known that a Co60

27

nucleus emits a β particle of energy 316 KeV followed by two γ emissions of energy 1173

and 1333 KeV respectively. Which of the following experimental methods would be the

best way to determine the lifetime of this Co6027 source?

(a) Measure the change in temperature of the source

(b) Measure the weight of the source now and again after one year

(c) Measure the recoil momentum of the nucleus during β emission

(d) Measure the number of γ photons emitted by this source

Q.7 A beam of hydrogen molecules travels in the z direction with a kinetic energy of 1 eV.

The molecules are in an excited state, from which they decay and dissociate into two

hydrogen atoms. When one of the dissociated atoms has its final velocity perpendicular to

the z direction, its kinetic energy is always 0.8 eV. The energy released in the dissociative

reaction is

(a) 0.26 eV (b) 2.6 eV (c) 0.36 eV (d) 3.6 eV





Q.8 Two parallel plates of metal sandwich a dielectric pad of thickness d, forming an ideal

capacitor of capacitance C. The dielectric pad is elastic, having a spring constant k. If an

ideal battery of voltage V across its terminals is connected to the two plates of this

capacitor, the fractional change 1dd in the gap between the plates is

(a) zero (b) 221

kd

CV

(c) 22

2

21

CVkd

CV

(d) 22

2

21

CVkd

CV

Q.9 When white light is scattered from a liquid, a strong absorption line is seen at 400 nm,

and two emission lines are observed, one of which is at 500 nm, and another in the infra-

red portion of the spectrum. The wavelength of this second emission line is

(a) 900 nm (b) 2000 nm (c) 100 nm (d) 222 nm

Q10 A detector is used to count the number of γ rays emitted by a radioactive source. If the

number of counts recorded in exactly 20 seconds is 10000, the error in the counting rate

per second is

(a) ± 5.0 (b) ± 22.4 (c) ± 44.7 (d) ± 220.0

Q.11 Consider a standard chess board with 8x8 squares. A piece starts from the lower left

corner, which we shall call Square (1, 1). A single move of this piece corresponds to

either one step right, i.e. to Square (1, 2) or one step forwards, i.e. to Square (2, 1). If it

continues to move according to these rules, the number of different paths by which the

piece can reach the Square (5, 5) starting from the Square (1, 1) is

(a) 120 (b) 72 (c) 70 (d) 45





Q.12 The uppermost graph in the set below shows the variation of current v/s voltage applied

across a copper conductor at temperature T1. Which of the graphs below – marked (a),

(b), (c) or (d) – will show the possible variation of the I–V curve for the same conductor

at another temperature T2 > T1?

(a) (b)

(c) (d)

2

1

0 1 2 3 4 5 V

2

1

0 1 2 3 4 5 V

2

1

0 1 2 3 4 5 V

2

1

0 1 2 3 4 5 V

2

1

0 1 2 3 4 5 V





Q.13 A ray of light is incident on a right-angled prism as shown in the figure below. The lower

surface of this prism is coated with a gel. If the incident ray makes angles (marked in

degrees) as shown in the figure, the refractive index of the gel must be

(a) 1.40 (b) 1.46 (c) 1.50 (d) 1.52

Q.14 A particle P1 is confined in a one-dimensional infinite potential well with walls at x = ±1.

Another particle P2 is confined in a one-dimensional infinite potential well with walls at

x = 0, 1. Comparing the two particles, one can conlude that

(a) the no. of nodes in the nth excited state of P1 is twice that of P2

(b) the no. of nodes in the nth excited state of P1 is half that of P2

(c) the energy of the nth level of P1 is the same as that of P2

(d) the energy of the nth level of P1 is one quarter of that of P2

Q.15 A charged particle is in the ground state of a one-dimensional harmonic oscillator

potential, generated by electrical means. If the power is suddenly switched off, so that the

potential disappears, then, according to quantum mechanics,

(a) the particle will shoot out of the well and move out towards infinity in one of the two

possible directions

(b) the particle will stop oscillating and as time increases it may be found farther and

farther away from the centre of the well

(c) the particle will keep oscillating about the same mean position but with increasing

amplitude as time increases

(d) the particle will undergo a transition to one of the higher excited states of the

harmonic oscillator

4529 74 45

PrismAir Air

AirGel

Ray Ray





Q.16 The pV diagram given below represents a

(a) Carnot refrigerator (b) Carnot engine

(c) gas turbine refrigerator (d) gas turbine engine

Q.17 In the laboratory, four point charges +Q, -Q, +Q, -Q are placed at the four ends of a

horizontal square of side a, as shown in the figure below. The number of neutral points

(where the electric field vanishes) is

(a) ∞

(b) 4

(c) 1

(d) zero

Q.18 Coherent monochromatic light falling through a small aperture produced a Fraunhofer

diffraction pattern as shown below

V

CB

A

P

D

a

a

Q

Q Q

Q

a

a





By looking at this diffraction pattern carefully one can guess that the shape of the

aperture was

(a) (b)

(c) (d)

Q.19 In the circuit given below, a person measures 9.0 V across the battery, 3.0 V across the 2

MΏ resistor RA and 4.5 V across the unknown resistor RB, using an ordinary voltmeter

which has a finite input resistance r. Assuming that the battery has negligible internal

resistance, it follows that (i) the resistance RB and and (ii) the input resistance r of the

voltmeter are, in MΏ,

(a) RB = 3.0, r = 6.0

(b) RB = 2.5, r = 7.5

(c) RB = 4.0, r = 12.0

(d) RB = 4.5, r = 10.0

2AR

R





Q.20 A heavy mass m is suspended from two identical steel wires of length l, radius r and

Young’s modulas Y, as shown in the figure below. When the mass is pulled down by a

distance x (x << l) and released, it undergoes elastic oscillations in the vertical direction

with a time period

(a) 2/cos2

22

Y

mlr

(b) g

l 2/cos2

(c) 2

2Yr

ml

(d) Y

mglr 2

2

xm

l l





B SECTION: (8 x 5=40 Marks)

Q.1 The wave function ψ of a quantum mechanical system described by a Hamiltonian H

can be written as a linear combination of Φ1 and Φ2 which are the eigenfunction of H

with eigenvalues 1E and 2E respectively. At t = 0, the system is prepared in the state

210 53

54

and then allowed to evolve with time. The wavefunction at time

21/21 EEhT will be (accurate to within a phase)

(a) 21 53

54

(b) Φ1 (c) 21 53

54

(d) Φ2 (e) 21 54

53

(f) 21 54

53

Q.2 Light transmitted along an optical fibre incurs losses due to Rayleigh scattering from

inhomogeneities. If a fibre of given length transmits 50% of the monochromatic light

coupled into it at a wavelength of 1350 nm, the transmitted fraction for the same fibre at

1550 nm will be

(a) 55% (b) 57% (c) 62% (d) 67% (e) 74% (f) 87%

Q.3 A quantum system has three energy levels-0.12 eV, -0.2 eV and -0.44 eV respectively.

Three electrons are distributed among these levels. At a temperature of 1727o C the

system has total energy -0.68 eV. The free energy of the system is approximately

(a) + 1.5 eV (b) +0.3 eV (c) -0.1 eV

(d) -0.3 eV (e) -1.0 eV (f) -1.5 eV

Q.4. An atom is capable of existing in two states: a ground state of mass M and an excited

state of mass M + ∆. If the transition from the ground state to the excited state proceeds

by the absorption of a photon, the photon frequency in the laboratory frame (where the

atom is initially at rest ) is

(a) hc2 (b)

Mh

c2

12

(c) h

Mc2

(d)

Mh

c2

12

(e)

MhMc

21

2

(f)

MhMc

21

2





Q.5 A plot of the common-emitter characteristics of a silicon n-p-n transistor is shown below.

Given this information and assuming that there will be a 0.7 V drop across a forward

biased silicon p-n junction, the approximate value of the output voltage Vout for an input

voltage Vin = 2V in the adjacent circuit will be

(a) 4V (b) 6V (c) 8V

(d)10 V (e) 12V (f) 14V

Q.6 Measurement of the electric field (E) and the magnetic field (B) in a plane-polarized

electromagnetic wave in vacuum led to the following:

tB

zE

yE

xE

,0 and

tE

zB

yB

xB

,0

It follows that

(a) jBBiEE ˆ,ˆ and the wave was travelling along k

(b) iBBjEE ˆ,ˆ and the wave was travelling along k

(c) kBBjEE ˆ,ˆ and the wave was travelling along i

(d) iBBkEE ˆ,ˆ and the wave was travelling along j

(e) kBBiEE ˆ,ˆ and the wave was travelling along j

(f) the wave was travelling along k but direction of E and B are not uniquely defined

inV

k65R b

15VVCC

500R L

outV

30

25

20

15

10

5

00 2 4 6 8 10 12 14 16

A10ICurrent Base b

A20Ib

A30Ib

A40Ib

A50Ib

Col

lect

or C

urre

nt I c

(mA

)

Collector-to-Emitter Voltage VCE (V)





Q.7 A mass m travels in a straight line with velocity 0v perpendicular to a uniform stick of

mass m and length L, which is initially at rest. The distance from the centre of the stick to

the path of the travelling mass is h (see figure). Now the travelling mass m collides

elastically with the stick, and the centre of the stick and mass m are observed to move

with equal speed v after the collision. Assuming that the travelling mass m can be treated

as a point mass, and the moment of inertia of the stick about its center is12

2mLI , it

follows that the distance h must be

(a) 2L (b)

4L (c)

6L

(d) 3

L (e) 3L (f) zero

Q.8 The binding energy per nucleon for 235U is 7.6 MeV. The 235U nucleus undergoes fission

to produce two fragments, both having binding energy per nucleon 8.5 MeV. The energy

released, in Joules, from the complete fission of 1 Kg of 235U is, therefore,

(a) 8000 (b) 1035 (c) 450

(d) 20000 (e) 8.7 x 1013 (f) 5.0 x 108

m

m

v

vh

L

ov





GS-2011 (Physics)




Name: ________________________________ Ref. Code: ____________





sheet.







contains 8 questions, each with 6 options. The maximum marks are 60 for Section A plus

40 for Section B, totaling to 100. Marking shall be as follows:











minutes each; questions in Section B should take about 10 minutes each. Obviously,

some questions may take a little less time while others may require a little more.







purpose.








GS-2011 (Physics)

A SECTION: 20 x 3= 60 Marks

Q.1 The infinite series

.....753

753

xxxx

where -1 < x < +1, can be summed to the value

(a) xx 1/1ln21 (b) xx 1/1ln

21

(c) tanhx (d)

x1tan41ln

Q.2 A 100 page book is known to have 200 printing errors distributed randomly through the

pages. The probability that one of the pages will be found to be completely free of errors

is closest to

(a) 13% (b) 25% (c) 50% (d) 67%

Q.3 Consider the matrix

010

100001

M

A 3-dimensional basis formed by eigenvectors of M is

(a)

110

and1

11

,1

12

(b)

110

110

,000

and

(c)

110

110

,1

11

and (d)

11

1

110

,1

12

and

Q.4 Two solid spheres S1 and S2 of the same uniform density fall from rest under gravity in a

viscous medium and, after some time, reach terminal velocities 1v and 2v respectively. If

the masses of S1 and S2 are 1m and 2m respectively, and 21 4vv , then the ratio 2

1m

m is

(a) 1/8 (b) 8 (c) ¼ (d) 4





Q.5 The dynamics of a particle of mass m is described in terms of three generalized

coordinates ζ, η and φ. If the Lagrangian of the system is

222

2

2

2

814

81

kmL

where k is a constant, then a conserved quantity in the system will be

(a) km (b) 22 // m

(c) 2222 // m (d) m

Q.6 A scientist is given two heavy spheres made of the same metal, which have the same

diameter and weight, and is asked to distinguish the spheres, without damaging them in

any way. Though the spheres look identical, one of them is actually a hallow spherical

shell, while the other is a set of concentric shells mounted on four thin rods of the same

metal (see figure).

To make this distinction, the scientist must perform an experiment where each sphere is

(a) immersed fully in a non-corrosive liquid and then weighed

(b) made into the bob of a long simple pendulum and set oscillating

(c) set rotating under the action of a constant torque

(d) given the same electric charge Q and the potential is measured

Q.7 A narrow beam of light of wavelength 589.3 nm from a sodium lamp is incident normally

on a diffraction grating of transmission type. If the grating constant is 1000000 m-1 the

number of principal maxima observed in the transmitted light will be

(a) 1 (b) 3 (c) 5 (d) 7





Q.8 A closed, thermally-insulated box contains one mole of an ideal monatomic gas G in

thermodynamic equilibrium with blackbody radiation B. The total internal energy of the

system is U = UG + UB where UG and UB (T4) are the energies of the ideal gas and the

radiation respectively. If UG = UB at the temperature T0 K, then the energy required to

raise the temperature from T0 K to ( T0 + 1) K, in terms of the gas constant R, is

(a) 0.33R (b) 1.5 R (c) 6R (d) 7.5R

Q.9 The phase diagram of a pure substance is given in the figure below, where ‘T’ denotes

the triple point and ‘C’ denotes the critical point.

The phase transitions occurring along the lines marked α, β and γ are

(a) α = melting; β = condensation; γ = sublimation

(b) α = sublimation; β = vaporisation; γ = melting

(c) α = melting; β = vaporisation; γ = condensation

(d) α = sublimation; β = melting; γ = vaporisation

Tem

pera

ture

Pressure

T

C





Q.10 The current read by the ammeter (A) in the circuit given below is

(a) 50.0 mA (b) 100.0 mA (c) 27.3 mA (d) 54.5 mA

Q.11 The sign of the majority charge carries in a doped silicon crystal is to be determined

experimentally. In addition to a voltage supply the combination of instruments needed to

perform the experiment is

(a) Thermometer, Voltmeter and Ammeter

(b) Magnet, Voltmeter and Ammeter

(c) pickup Coil, Voltmeter and Ammeter

(d) Heater, Magnet and Thermometer

Q.12 A small but very powerful bar magnet falls from rest under gravity through the centre of

a horizontal ring of conducting wire, as shown in the figure below (on the left). The

speed-versus-time graph, in arbitrary units of the magnet will correspond most closely to

which of the four plots below (on the right)?

100

100

100

100100

5 5V 6





Q.13 The spectra of electromagnetic radiation emitted by distant objects like stars and galaxies

give important clues about their physical properties. In this context, a correct statement is

that

(a) the wavelengths in the emission spectrum of an element in a star are always the same

as those found in laboratory experiments

(b) absorption spectra cannot be used to determine which molecules are present in the

distant objects

(c) the nuclear structure of the distant objects cannot be determined from lines in the

visible region of the spectrum

(d) absorption lines in the spectra of distant objects do not carry information about their

motion in a direction transverse to the line of sight

Q14 Given that the ionization energies of Hydrogen (1H) and Lithium (3Li) are 13.6 eV and

5.39 eV, respectively, the effective nuclear charge experience by the valence electron of a 3Li atom may be estimated in terms of the proton charge e as

(a) 3.00 e (b) 1.59 e (c) 1.26 e (d) 0.63 e

0

(a)

Time

Spee

d

0

(c)

Time

Spee

d 0

Time

Spee

d (b)

0Time

Spee

d (d)





Q.15 Two identical non-interacting particles, each of mass m and spin21 , are placed in a one-

dimensional box of length L. In quantum mechanics, the lowest possible value of the total

energy of these two particles is 0 . If instead, four such particles are introduced into a

similar one-dimensional box of length 2L, then the lowest possible value of their total

energy will be

(a) 0 (b) 45 0 (c) 4

3 0 (d) 02

Q.16 An excited atomic electron undergoes a spontaneous transition

2/12/3 23 pd

The interaction responsible for this transition must be of the type

(a) electric dipole (E1) OR magnetic dipole (M1)

(b) electric quadrupole (E2) OR magnetic dipole (M1)

(c) electric quadrupole (E2) OR magnetic quadrupole (M2)

(d) electric dipole (E1) OR magnetic quadrupole (M2)

Q.17 A fast-moving 14N nucleus collides with an α particle at rest in the laboratory frame,

giving rise to the reaction

14N + α → 17O + p

Given the masses 14.00307 a.m.u and 16.99913 a.m.u. for 14N and 17O nuclei

respectively, and 4.00260 a.m.u. and 1.00783 a.m.u. for α and p respectively, the

minimum kinetic energy in the laboratory frame of the 14N nucleus must be

(a) 1.20 MeV (b) 1.55 MeV (c) 4.20 MeV (d) 5.41 MeV





Q.18 An unmagnetised sample of iron is placed in a magnetic field H which varies with time as

shown in the plot below.

The magnetization M of the iron sample is continuously measured and also plotted as a

function of time. The appearance of this plot will be closest to

(a) (b)

(c) (d)

0 2 4 6 8

6.03.00.03.06.0

Time (min)

H (k

ilo o

erst

ed)

1

0

0 2 4 6 8

1

Time (min)

M (a

rb. u

nits

)

1

0

0 2 4 6 8

1

Time (min)

M (a

rb. u

nits

)

1

0

0 2 4 6 8

1

Time (min)

M (a

rb. u

nits

)

1

0

0 2 4 6 8

1

Time (min)

M (a

rb. u

nits

)





Q.19 The figure below shows the Bragg diffraction pattern for X-rays of wavelength 1.54 Å

incident on two crystalline Silicon thin film Sample A and B. The dashed line

corresponds to a normal Sample A and the continuous line corresponds to another

Sample B, which is modified due to differences in the growth conditions.

These plots suggest that the modified sample B is

(a) stretched in all direction by 3%

(b) compressed in all directions by 3%

(c) stretched in the z direction by 1% and possibly compressed in x and y directions

(d) compressed in the z direction by 1% and possibly stretched out in x and y directions

(A) normal

(B) modified

2928 30Diff

ract

ed X

-ray

inte

nsity

(arb

. Uni

ts)

Sample Si2

zo

ray-X A5.1

(deg)2





Q.20 The digital electronic circuit shown below (left side) has some problem and is not

performing as intended. The voltage at each pin as a function of time is shown in the

adjacent figures.

The problem in the about circuit may be that

(a) the Pin 8 is clamped to +5 (b) the input inverter is shorted

(c) the Pin 6 is shorted to ground (d) OR gate is used instead of AND gate

V5

01Pin

V5

04Pin

V5

05Pin

V5

06Pin

V5

08Pin

8965421

Y





GS-2011 (PHYSICS)

PART– B (MARKS: 8 x 5=40)

Q.1 The trace of the real 4 x 4 matrix U = exp(A), where

0004/004/004/004/000

A is equal to

(a) 2 (b) π/4 (c) exp(iφ) for φ = 0, π

(d) zero (e) π/2 (f) 22

Q.2 A region of space is divided into two parts by a plane P, as shown in the figure below. A

particle of mass m passes from Region I and Region II where it has speed 1v and 2v

respectively. There is a constant potential U1 in Region I and U2 in Region II.

Let T1 be the kinetic energy of the particle in Region I. If the trajectory of the particle is

inclined to the normal to the plane P by angles θ1 and θ2, as shown in the figure then the

ratio sin θ1/ sin θ2 is given by

(a) 211 /1 UUT (b) 211 /1 UUT (c) 121 /1 TUU

(d) 121 /1 TUU (e) 2111 UUT (f) 121 /1 TUU

P

1V

2V

2

1

IRegion IIRegion

2U1U





Q.3 The electric field of an electromagnetic wave of angular frequency ω propagating in a

medium with conductivity σ, permittivity and permeability µ is given by

kxtiEE exp0

where the imaginary part of the complex propagation constant k is 2/ multiplied

by the factor

(a) 2/1

1/1

(b) 2/1

1/1

(c) 2/1

2 1/1

(d)

2/12 1/1

(e) 2/1

1/1

(f) 2/1

1/1

Q.4 A system having N non-degenerate energy eigenstates is populated by N identical spin-

zero particles and 2N identical spin-half particles. There are no interactions between any

of these particles. If N = 1000, the entropy of the system is closet to

(a) zero (b) 1000 kB (c) 13.82 kB

(d) 693.1 kB (e) 5909.693 kB (f) 6909 kB

Q.5 The Michelson interferometer in the figure below can be used to study properties of light

emitted by distant sources.

Source

Mirror Fixed

Splitter Beam

Detector Photo

Mirror Movable





A Source 1S , when at rest, is known to emit light of wavelength 632.8 nm. In this case, if

the Movable Mirror is translated through a distance d, it is seen that 99,565 interference

fringes pass across the Photo-Director. For another source S2, moving at an uniform

speed of 1.5 x 107 ms-1 towards the interferometer along the straight line joining it to the

Beam Splitter, one sees 100,068 interference fringes pass across the Photo-Detector for

the same displacement d of the Movable Mirror. It follows that S2, in its own rest frame,

must be emitting light of wavelength

(a) 599.6 nm (b) 598.9 nm (c) 628.0 nm

(d) 631.2 nm (e) 661.9 nm (f) 662.8 nm

Q.6 A particle of mass m is placed in the ground state of a one-dimensional harmonic

oscillator potential of the form

2

21 kxxV

where the stiffness constant k can be varied externally. The ground state wavefunction

has the form kaxx 2exp where a is a constant. If, suddenly, the parameter k is

changed to 4k, the probability that the particle will remain in the ground state of the new

potential is

(a) 0.06 (b) 0.94 (c) 0.53

(d) 0.47 (e) 0.33 (f) 0.67

Q.7 A Cloud Chamber of width 0.01 m is filled with pure nitrogen gas (N2) at normal

temperature and pressure. A beam of α particles, when incident normally on the chamber,

make tracks which are visible under strong illumination. Whenever an α particle He42 has

a nuclear collision with a N147 nucleus, the track shows a distinct bend. The radius of a

nucleus is given by 3/10 Arr where 0r = 1.27 x 10-5 m and A is the atomic mass number.

If the α particles move at non-relativistic speeds and the total number of incident of α

particles is 107, the number of such distinct bends is approximately

(a) 100 (b) 200 (c) 300

(d) 400 (e) 500 (f) 600





Q.8 The three electronic circuits marked (i), (ii) and (iii) in the figure below can all work as

logic gates, where the input signals are either 0V or 5V and the output is Vo.

Identify the correct combination of logic gates (i), (ii), (iii) in the options given below.

(a) AND, OR, NOR (b) OR, NAND, NOR (c) NAND, AND, XOR

(d) XOR, AND, NAND (e) NOR, XOR, AND (f) NOR, NAND, OR

oV

(ii)

oV

V(iii)(i)

AB

oV

V





GS-2012 (Physics)




Name: ________________________________ Ref. Code: ____________





sheet.







contains 10 questions, each with 5 options. The maximum marks are 60 for Section A

plus 40 for Section B, totaling to 100. Marking shall be as follows:











minutes each; questions in Section B should take about 8 minutes each. Obviously, some

questions may take a little less time while others may require a little more.







purpose.








GS-2012 (Physics) A SECTION: 20 x 3= 60 Marks

A1. Two different 2 x 2 matrices A and B are found to have the same eigenvlues. It is then

correct to state that A = SBS-1 where S can be a

(a) traceless 2 x 2 matrix (b) Hermitian 2 x 2 matrix

(c) unitary 2 x 2 matrix (d) arbitrary 2 x 2 matrix

A2. The function f(x) represents the nearest integer less than x, e.g.

f (3.14) = 3 .

The derivative of this function (for arbitrary x) will be given in terms of the integers n as

xf =

(a) 0 (b) n

nx (c) n

nx (d) n

nxf

A3. Two masses M1 and M2 (M1 < M2) are suspended from a perfectly rigid horizontal

support by a system of three taut massless wires W1, W2 and W3, as shown in the figure.

All the three wires have identical cross-sections and elastic properties and are known to

be very strong.

If the mass M2 is increased gradually, but without limit, we should expect the wires to

break in the following order:

(a) first W2, then W1 (b) first W1, then W2

(c) first W2, then W3 (d) first W3

1W2W

o60o30

1M

2M

3W





A4. A high-velocity missile, travelling in a horizontal line with a kinetic energy of 3.0 Giga-

Joules (GJ), explodes in flight and breaks into two pieces A and B of equal mass. One of

these pieces (A) flies off in a straight line perpendicular to the original direction in which

the missile was moving and its kinetic energy is found to be 2.0 GJ. If gravity can be

neglected for such high-velocity projectiles, it follows that the other piece (B) flew off in

a direction at an angle with the original direction of

(a) 30o (b) 42330 (c) 45o (d) 60o

A5. Consider a spherical planet, rotating about an axis passing through its centre. The

velocity of a point on its equator is eqv . If the acceleration due to gravity g measured at

the equator is half of the value of g measured at one of the poles, then the escape velocity

for a particle shot upwards from that pole will be

(a) υeq/2 (b) υeq/√2 (c) √2 υeq (d) 2 υeq

A6. A dynamical system with two degrees of freedom, has generalized coordinates 1q and

2q , and kinetic energy

21qqT

If the potential energy is 0, 21 qqV , the correct form of the Hamiltonian for this

system is

(a) /21 pp (b) 21qq

(c) 2/2211 qpqp (d) 2/1221 qpqp





A7. An ideal liquid of density 1 gm/cc is flowing at a rate of 10 gm/s through a tube with

varying cross-section, as shown in the figure.

Two pressure gauges attached at the points A and B (see figure) show readings of PA and

PB respectively. If the radius of the tube at the points A and B is 0.2 cm and 1.0 cm

respectively, then the difference in pressure (PB − PA), in units of dyne cm-2, is closest to

(a) 100 (b) 120 (c) 140 (d) 160

A8. Unpolarised light of intensity I0 passes successively through two identical linear

polarisers A and B, placed such that their polarisation axes are at an angle of 45o (see

figure) with respect to one another.

Assuming A and B to be perfect polarisers (i.e. no absorption losses), the intensity of the

transmitted light will be IT =

(a) 4/0I (b) 22/0I (c) 2/0I (d) 2/0I

A

B

o45

A B

TI0I





A9. Three equal charges Q are successively brought from infinity and each is placed at one

of the three vertices of an equilateral triangle. Assuming the rest of the Universe as a

whole to be neutral, the energy E0 of the electrostatic field will increase, successively,

to

E0 + Δ1, E0 + Δ1 + Δ2, E0 + Δ1 + Δ2 + Δ3

Where Δ1 : Δ2 : Δ3 =

(a) 1 : 2 : 3 (b) 1 : 1 : 1 (c) 0 : 1 : 1 (d) 0 : 1 : 2

A10. Five sides of a hollow metallic cube are grounded and the sixth side is insulated from the

rest and is held at a potential Φ (see figure).

The potential at the center O of the cube is

(a) 0 (b) Φ/6 (c) Φ/5 (d) 2Φ/3

A11. Consider a sealed but thermally conducting container of total volume V, which is in

equilibrium with a thermal bath at temperature T. The container is divided into two equal

chambers by a thin but impermeable partition. One of these chambers contains an ideal

gas, while the other half is a vacuum (see figure).

O





If the partition is removed and the ideal gas is allowed to expand and fill the entire

container, then the entropy per molecule of the system will increase by an amount

(a) Bk2 (b) 2/1lnBk (c) 2lnBk (d) 2/2lnBk

A12. When a gas is enclosed in an impermeable box and heated to a high temperature T, some

of the neutral atoms lose an electron and become ions. If the number density of neutral

atoms, ions and electrons is Na, Ni and Ne, respectively, these can be related to the

average volume Va occupied by an atom/ion and the ionisation energy E by the relation

(a) Ne (Na + Ni) = (Na/Va) exp (−E/ Bk T)

(b) Na (Ne + Ni) = (Na/Va) exp (−E/ Bk T)

(c) NeNi = (Na/Va) exp (+E/ Bk T)

(d) NeNi = (Na/Va) exp (−E/ Bk T)

A13. In a scanning tunnelling microscope, a fine Platinum needle is held close to a metallic

surface in vacuum and electrons are allowed to tunnel across the tiny gap δ between the

surface and the needle. The tunnelling current I is related to the gap δ, through positive

constants a and b, as

(a) I = a − bδ (b) I = a + bδ

(c) log I = a − bδ (d) log I = a + bδ

gas vacuum





A14. A particle in a one-dimensional potential has the wavefunction

ax

ax exp1

where a is a constant. It follows that for a positive constant V0, the potential V(x) =

(a) V0 x2 (b) V0 |x| (c) −V0 δ(x) (d) −V0/|x|

A15. Consider the high excited states of a Hydrogen atom corresponding to large values of the

principal quantum number (n >>1). The wavelength λ of a photon emitted due to an

electron undergoing a transition between two such states with consecutive values of n

(i.e. nn 1 ) is related to the wavelength λα of the Kα line of Hydrogen by

(a) λ = n3 λα/8 (b) λ = 3n3λα/8 (c) λ = n2 λα (d) λ = 4λα /n2

A16. A proton is accelerated to a high energy E and shot at a nucleus of Oxygen O168 . In

order to penetrate the Coulomb barrier and reach the surface of the Oxygen nucleus,

E must be at least

(a) 3.6 MeV (b) 1.8 MeV (c) 45 keV (d) 180 eV

A17. A monochromatic beam of X-rays with

wavelength λ is incident at an angle θ on a

crystal with lattice spacings a and b as

sketched in the figure below.

A condition for there to be a maximum

in the diffracted X-ray intensity is

(a) sin2 22 ba

(b) 2b cos θ = λ

(c) 2 a cos θ = λ

(d) (a + b) sin θ = λ

b

a





A18. Suppose the energy band diagram of a certain pure crystalline solid is as shown in the

figure below, where the energy (E) varies with crystal momentum (k) as E k2. .

At finite temperatures the bottom of the conduction band (CB) is partially filled with

electrons (e) and the top of the valence band (VB) is partially filled with holes (h). If an

electric field is applied to this solid, both e and h will start moving. If the time between

collisions is the same for both e and h, then

(a) e and h will move with the same speed in opposite directions

(b) h will on an average achieve higher speed than e

(c) e will on an average achieve higher speed than h

(d) e and h will recombine and after a while there will be no flow of charges

A19. Consider the circuit shown below.

The minimum number of NAND gates required to design this circuit is

(a) 6 (b) 5 (c) 4 (d) 3

A

BY

Crystal momentum k

Ener

gy

CB

VB

ok





A20. Consider the following circuit:

If the waveform given below is fed in at Vin,

then the waveform at the output Vout will be

(a) (b)

(c) (d)

inVC

R

outV

inV

pV

pV

outV outV

outV outV





GS – 2012(PHYSIC)

B-Section (Marks 10x4=40)

B1. Consider the integral

2

2 22

p

p pxdx

where p is a constant. This integral has a real, nonsingular value if

(a) p < −1 (b) p > 1 (c) p = 1 (d) p → 0 (e) p → ∞

B2. If we model the electron as a uniform sphere of radius er , spinning uniformly about an

axis passing through its centre with angular momentum 2/eL , and demand that the

velocity of rotation at the equator cannot exceed the velocity c of light in vacuum, then

the minimum value of er is

(a) 19.2 fm (b) 0.192 fm (c) 4.8 fm (d) 1960 fm (e) 480 fm

B3. The intensity of light coming from a distant star is measured using two identical

instruments A and B, where A is placed in a satellite outside the Earth’s atmosphere, and

B is placed on the Earth’s surface. The results are as follows:

Colour Wavelength (nm)

Intensity at A (nanoWatts)

Intensity at B (nanoWatts)

green 500 100 50

red 700 200 x

Assuming that there is scattering, but no absorption of light in the Earth’s atmosphere at

these wavelengths, the value of x can be estimated as

(a) 137 (b) 147 (c) 157 (d) 167 (e) 177





B4. Consider three identical infinite straight wires A, B and C arranged in parallel on a plane

as shown in the figure.

The wires carry equal currents I with directions as shown in the figure and have mass per

unit length m. If the wires A and C are held fixed and the wire B is displaced by a small

distance x from its position, then it (B) will execute simple harmonic motion with a time

period

(a)

Idm

0

2

(b)

Idm

0

22 (c)

Idm

0

2

(d)

Idm

022

(e)

Idm

0

2

B5. The normalized wavefunctions of a Hydrogen atom are denoted by xmln ,, , where n, l

and m are, respectively, the principal, azimuthal and magnetic quantum numbers

respectively. Now consider an electron in the mixed state

xxxx 2,2,30,1,20,0,1 32

32

31

The expectation value E of the energy of this electron, in electron-Volts (eV) will be

approximately

(a) −1.5 (b) −3.7 (c) −13.6 (d) −80.1 (e) +13.6

d d

I

I

Ix

A B C





B6. The strongest three lines in the emission spectrum of an interstellar gas cloud are found to

have wavelengths 0 , 02 and 06 respectively, where 0 is a known wavelength.

From this we can deduce that the radiating particles in the cloud behave like

(a) free particles (b) particles in a box (c) harmonic oscillators

(d) rigid rotators (e) hydrogenic atoms

B7. When light is emitted from a gas of excited atoms, the lines in the spectrum are Doppler-

broadened due to the thermal motion of the emitting atoms.

The Doppler width of an emission line of wavelength 500 nanometres (nm) emitted by

an excited atom of Argon A4020 at room temperature (27o C) can be estimated as

(a) 5.8 × 10−4 nm (b) 3.2 × 10−4 nm (c) 3.2 × 10−3 nm

(d) 2.5 × 10−3 nm (e) 1.4 × 10−3 nm

B8. In a nuclear reactor, Plutonium Pu23994 is used as fuel, releasing energy by its fission into

isotopes of Barium Ba14654 and Strontium Sr91

38 through the reaction

n3SrBanPu 10

9138

14656

10

23994

The binding energy (B.E.) per nucleon of each of these nuclides is given in the table

below:

Nuclide Pu23994 Ba146

54 Sr9138

B.E. per nucleon (MeV) 7.6 8.2 8.6

Using this information, one can estimate the number of such fission reactions per second

in a 100 MW reactor as

(a) 3.9 × 1018 (b) 7.8 × 1018 (c) 5.2 × 1019

(d) 5.2 × 1018 (e) 8.9 × 1017





B9. Metallic Copper is known to form cubic crystals and the lattice constant is measured from

X-ray diffraction studies to be about 0.36 nm. If the specific gravity of Copper is 8.96 and

its atomic weight is 63.5, one can conclude that

(a) the crystals are of simple cubic type

(b) the crystals are of b.c.c. type

(c) the crystals are of f.c.c. type

(d) the crystals are a mixture of f.c.c. and b.c.c. types

(e) there is insufficient data to distinguish between the previous options

B10. The voltage regulator circuit shown in the figure has been made with a Zener diode rated

at 15V, 200mW. It is required that the circuit should dissipate 150mW power across the

fixed load resistor RL.

For stable operation of this circuit, the input voltage Vi must have a range

(a) 17.5 V – 20.5 V (b) 15.5 V – 20.5 V (c) 15.5 V – 22.5 V

(d) 17.5 V – 22.5 V (e) 15.0 V – 22.5 V

iV oVLR

238





GS-2012 (Physics)




Name: ________________________________ Ref. Code: ____________





sheet.




3. This test comes in two sections, Section A, Section B and Section C. Test paper

contain multiple choice-type questions. Only ONE of the options given at the end of each

question is correct. Section A contains 25 questions Section B and Section C contain 15

questions each. Marking shall be as follows:

(i) If the answer is correct: +3 marks

(ii) If the answer is incorrect: -1

(iii) If the answer is not attempted: 0 marks

(iv) If more than one box is marked: 0 marks







minutes each; questions in Section B and Section C should take about 8 minutes each.

Obviously, some questions may take a little less time while others may require a little

more.







purpose.








GS-2013-Y (Physics)

Section A

To be attempted by ALL candidates

Q1. The value of the integral

0

29 exp xdxx is

(a) 20160 (b) 24 (c) 18 (d) 12

Q2. A two-dimensional vector tA

is given by

tjtltA 3cosˆ2sinˆ

Which of the following graphs best describes the locus of the tip of the vector, as t is

varied from 0 to 2 ?

(a) (b)

(c) (d)

1

5.0

0

5.0

11 5.0 0 5.0 1

1

5.0

0

5.0

11 5.0 0 5.0 1

1

5.0

0

5.0

1

1 5.0 0 5.0 1

1

5.0

0

5.0

1

1 5.0 0 5.0 1

j

i





Q3. The differential equation

022

2

ydxdy

dxyd

has the complete solution, in terms of arbitrary constants A and B,

(a) A exp x + B exp (-x) (b) A exp x + Bx exp (-x)

(c) x{A exp x + B exp (-x)} (d) A exp x + Bx exp x

Q4. A stone is dropped vertically from the top of a tower of height 40m. At the same time

a gun is aimed directly at the stone from the ground at a horizontal distance 30m from the

base of the tower and fired. If the bullet from the gun is to hit the stone before it reaches

the ground, the minimum velocity of the bullet must be approximately.

(a) 57.4 ms-1 (b) 27.7 ms-1 (c) 17.7 ms-1 (d) 7.4 ms-1

Q5. Consider the uniform solid right cone depicted in the

figure on the right. This cone has mass M and a circular

base of radius r. If the moment of inertia of the cone about

an axis passing through the centre of mass OC.M.

(see figure) is given by

224803 hrM

then the moment of inertia about another axis parallel to the

X axis, but passing through the point O1 (see figure), is

(a) 224803 hrM (b) 22 223

201 hrM

(c) 222403 hrM (d) 22 415

301 hrM

..MCO

Z

rO 1O Y

h41

X

h





Q6. Two planets A and B move around the Sun in elliptic orbits with time periods TA and

TB respectively. If the eccentricity of the orbit of B is ε and its distance of closest

approach to the Sun is R, then the maximum possible distance between the planets is

[Eccentricity of an ellipse:minmax

minmax

rrrr

]

(a) RTT

B

A

2/3

2/3

2

2

111

(b) R

TT

B

A

3/2

3/2

2

2

111

(c) RTT

B

A

3

3

111

(d) R

TT

B

A

3/2

3/2

111

Q7. A spaceship S blasts off from the Earth. After some time, Earth station informs the

crew that they have settled into a constant velocity 0.28c radially outwards from the

Earth, but unfortunately they are on a head-on collision course with an asteroid A at a

distance of 15 light-minutes coming in towards the Earth along the same radius (see

figure below)

Instruments on-board the spaceship immediately estimate the speed of the asteroid to

have a constant value 0.24c. It follows that the maximum time (in minutes) available to

the crew to evacuate the ship before the collision is

(a) 29 (b) 30 (c) 60 (d) 63

Q8. A point charge q sits at a corner of a cube of side a, as shown in the

figure on the right. The flux of the electric field vector through the shaded

side (region A) is

(a) 024

q (b) 016

q (c) 08

q (d) 06

q

S Ad

vu

Earth

q A





Q9. A parallel plate capacitor of circular cross section with radius r >> d, where d is the

spacing between the plates, is charged to a potential V and then disconnected from the

charging circuit. If, now, the plates are slowly pulled apart (keeping them parallel) so that

their separation is increased from d to d', the work done will be

(a)

'1

2

220

dd

dVr

(b)

1'

2

220

dd

dVr

(c) dd

dVr '

2

220

(d) '2

220

dd

dVr

Q10. In the laboratory frame two electrons are shot at each other with equal and opposite

velocities 1u and 2u respectively, but not along the same straight line, as shown below.

Each electron will be acted on by the Coulomb repulsion due to the other, as well as the

Lorentz force due to its own motion in the magnetic field created by the other. Which of

the diagrams given below best describes the final velocities 1v and 2v of these electrons?

[You may assume that the electrons are distinguishable.]

(a) (b)

(c) (d)

1u

2ux

z

1v

2vz

x2v

1vz

x

2v

1v z

x

2v

1vz

x





Q11. A certain amount of fluid with heat capacity CF Joules/oC is initially at a

temperature 0o C. It is then brought into contact with a heat bath at a temperature of

100o C, and the system is allowed to come into equilibrium. In this process, the entropy

(in Joules/oC) of the Universe changes by

(a) 100 CF (b) 0.044 CF (c) 0.055 CF (d) 0

Q12. A monatomic gas is described by the equation of state

nRTbnVp

where b and R are constants and other quantities have their usual meanings. The

maximum density (in moles per unit volume) to which this gas can be compressed is

(a) b1 (b) b (c)

bn1 (d) infinity

Q13. In a quantum mechanical system, an observable A is represented by an operator .A

If is a state of the system, but not an eigenstate of A , then the quantity

22 ˆˆ AAr

satisfies the relation

(a) r > 0 (b) r = 0 (c) r < 0 (d) r ≤ 0

Q14. Consider a quantum mechanical system with three linear operators ,A B and

,C which are related by

ICBA ˆˆˆˆ

where I is he unit operator. If dxdA ˆ and ,ˆ xB then C must be

(a) zero (b) dxd (c)

dxdx (d)

dxdx





Q15. A particle of energy E moves in one dimension under the influence of a potential

V(x). If E > V(x) for some range of x, which of the following graphs can represent a

bound state wave function of the particle?

(a) (c)

(b) (d)

Q16. In a Davisson–Germer experiment, a collimated beam of electrons of energy 54 eV,

at normal incidence on a given crystal, shows a peak at a reflection angle of 40o. If the

electron beam is replaced by a neutron beam, and the peak appears at the same value of

reflection angle, then the energy of the neutrons must be

(a) 33 eV (b) 0.03 eV (c) 0.3 eV (d) 330 eV

Q17. The velocity of an electron in the ground state of a hydrogen atom is Hv . If vp be the

velocity of an electron in the ground state of positronium, then

(a) Hp vv (b) Hp vv 2 (c) 2H

pvv (d) Hp vv 2

x

x

x

x

x

x

x

x





Q18. Consider an ensemble of microscopic quantum mechanical systems with two energy

levels 1E and 2E , where 1E < 2E . Which of the following graphs best describes the

temperature dependence of the average energy E of the system?

(a) (b)

(c) (d)

Q19. A ray of light is incident on the surface of a thin

prism at a small angle θ1 with the normal, as shown in

the figure on the right. The material of the prism has

refractive index n and you may assume the outside

refractive index to be unity. If the (small) apex angle of

the prism is α, the deviation angle δ (angle between the

incident and exited ray; see figure) is given by

(a) α (b) αn (c) α(n – 1) (d) α(n + 1)

E 212

1 EE

1E

T

E 212

1 EE

1E

T

E 212

1 EE

1E

T

E 212

1 EE

1E

T

n

1





Q20. A parallel beam of light of wavelength λ is incident on a

transmission grating with groove spacing d, at an angle θi, as

shown in the figure. The plane of incidence is normal to the

grooves. After diffraction, the transmitted beam is seen to be at

an angle α relative to the normal. Which of the following

conditions must be satisfied for this to happen?

(a) nd i sinsin (b) nd i sinsin

(c) nd i sin2 (d) nd i sin2

Q21. Let EN be the energy released when one mole of pure 235U undergoes controlled

fission, and EC be the energy released when one mole of pure carbon undergoes

complete combustion. The ratio EN /EC will have the order of magnitude

(a) 104 (b) 108 (c) 109 (d) 106

Q22. The entropy S of a black hole is known to be of the form

S = αkBA

where A is the surface area of the black hole and α is a constant, which can be written in

terms of c (velocity of light in vacuum), ħ (reduced Plank’s constant) and GN (Newton’s

constant of gravitation). Taking the radius of the black hole as

2

2c

MGR N

it follows that the entropy S is

[λ is a numerical constant]

(a) 4

22

ckMG BN

(b)

MGck

N

B

(c) 4

22

ckMG BN

(d)

ckMG BN

2

d

i

Grating





Q23. The circuit depicted on the right has been made with a

silicon n-p-n transistor.

Assuming that there will be a 0.7 V drop across a forward-

biased silicon p-n junction, the power dissipated across the

transistor will be, approximately,

(a) 53 mW

(b) 94 mW

(c) 17 mW

(d) 67 mW

Q24. An input of 1.0 V DC is given to the ideal Op-Amp circuit depicted below. What

will be the output voltage?

(a) 10.0 V (b) 1.0 V (c) 0 V (d) – 9.0 V

Q25. The circuit below uses only NAND gates. Find the final output.

(a) A XOR B (b) A OR B (c) A AND B (d) A NOR B

VVCC 15

500LR

100R

outV

VVin 2

k10'R

Amp Op

1mHL F1C k90R

outVinV

AB





GS – 2013 – Y (Physics)

Section B

To be attempted only by candidates for Integrated Ph.D. programme.

(Candidate for Ph.D. programme will get no credit for attempting this section.)

Q26. Consider the surface corresponding to the equation

4x2 + y2 + z = 0

A possible unit tangent to this surface at the point (1, 2, -8) is

(a) kji ˆ5

4ˆ5

3ˆ5

1 (b) kj ˆ

54ˆ

51

(c) kji ˆ91ˆ

98ˆ

94

(d) ji ˆ5

2ˆ5

1

Q27. A particle with time-varying mass ,10

tmtm where m0 and τ are positive

constants, moves along the x-axis under the action of a constant positive force F for

0 ≤ t < τ. If the particle is at rest at time t = 0, then at time t = t, its velocity v will be

(a)

tmF 1log

0

(b) t

mFt log

0

(c) 1

0

1

t

mFt (d)

tmF 1

0

Q28. A ball of mass m slides under gravity without

friction inside a semicircular depression of radius

a inside a fixed block placed on a horizontal surface, as

shown in the figure. The equation of motion of the ball

in the x-direction will be

(a) 2

2

1axx

agx (b) 2

2

1axx

agx

(c) xagx (d) x

agx

z

x





Q29. A particle P, of rest mass M and energy E, suddenly decays into two particles A and

B of rest masses mA and mB respectively, and both particles move along the straight line

in which P was moving. A possible energy EA of the particle A will be

(a)

2

12 M

mmE BA (b)

2

2

22

12 M

mmE BA

(c)

2

12 M

mmE BA (d)

2

22

12 M

mmE BA

Q30. Consider two charges +Q and –Q placed at the points

(a, 0) and (-a, 0) in a place, as shown in the figure on the

right. If the origin is moved to the point (X, Y), the

magnitude of the dipole moment of the given charge

distribution with respect to this origin will be

(a) 2222 yXaQyXaQ (b) 2Qa

(c) XaQXaQ (d) 222 YXQa

Q31. A plane electromagnetic wave traveling in a vacuum is characterized by the electric

and magnetic fields

kztiVmiE exp30ˆ 1

kztiAmHjH expˆ 10

If ω, k > 0, the value of H0 must be

(a) 0.25 (b) 0.67 (c) 0.94 (d) 2 π

Q32. Which of the following classic experiments provides unambiguous proof that the

Earth is a non-inertial frame of reference with respect to the fixed stars?

(a) Fizeau’s rotating wheel experiment (b) Michelson-Morley experiment

(c) Foucault’s pendulum experiment (d) Newton’s coin-and-feather experiment

Q

0,a 0,a

Q

YX ,





Q33. A cross-shaped opening is illuminated by a parallel beam of white light. A thin

plano-convex cylindrical glass lens is placed 20 cm in front of it, as shown in the figure

below.

The radius of curvature of the curved surface of the lens is 5 cm and 1.5 is the refractive

index of glass. On a screen placed as shown at the plane where a real image forms on the

other side of the lens, the image of the opening will appear as

(a) (b)

(c) (d)

screen cm20

x

yz





Q34. A classical ideal gas, consisting of N particle (N → ∞) is confined in a box of

volume V at temperature T and pressure p. The probability that, at any instant of time, a

small sub-volume v0 becomes totally void (i.e. no particles inside), due to a spontaneous

statistical fluctuation, is

(a)

Vv0exp (b)

NTpv0 (c)

NTpV

Vv

exp0 (d)

V

Nv0exp

Q35. The pV diagram for a Carnot cycle executed by an ideal gas with CP/CV = γ > 1 is

shown below. Note that 1, 2, 3 and 4 label the change over points in the cycle

If, for this cycle,

X

pp

TT

3

2

3

2

then X =

(a) 11 (b) 0 (c) 1 (d)

1

Q36. A harmonic oscillator has the wave function,

txtxtxtx ,22,22,351, 210

where txn , is the eigenfunction belonging to the nth energy eigenvalue

21n .

The expectation value E of energy for the state tx, is

(a) 1.58 ħω (b) 0.46 ħω (c) ħω (d) 1.46 ħω

p

12

34

V





Q37. An energy eigenstate of the Hydrogen atom has the wave function

i

ar

armn

0

2/3

0 3expcossin1

811,,

where a0 is the Bohr radius. The principal (n), azimuthal (ℓ ) and magnetic (m) quantum

numbers corresponding to this wave function are

(a) n = 3, ℓ = 2, m = 1 (b) n = 2, ℓ = 1, m = 1

(c) n = 3, ℓ = 2, m = -1 (d) n = 2, ℓ = 1, m = ±1

Q38. The Curie temperature of a

single crystal of PrGe is known to be

41 K. The magnetization data of this

sample is measured at 1.8 K for the

magnetic field applied parallel to the

[001] direction is shown in the figure

on the right. At a temperature of

38 K, the hysteresis loop in the figure

will

(a) increase in width

(b) decrease in width

(c) have the same width

(d) shrink to a line

Q39. Consider two energies of a free electron gas in a metal at an absolute temperature T,

viz.,

E± = EF ± ∆

where EF is the Fermi level. If the corresponding electron populations n(E±) satisfy the

relation n(E-) / n(E+) = 2, then ∆ =

(a) kBT (b) kBT ln 2 (c) 2TkB (d) 2 kBT

10 5 0 5 104

2

0

2

4

]001//[8.1

HKT

GePr

Field(kOe)

Mag

netiz

atio

n (µ

B/P

r)





Q40. The figure below shows the current voltage characteristics of a diode over a range

of voltage and current where it is safe to operate the diode.

When this diode is used in the circuit on the extreme right, the approximate current, in

mA, through the diode will be

(a) 8.3 (b) 16.7 (c) 25 (d) 0

40

20

0

20

40

10 9 8 7 1 2 3 4

Voltage (V)

Cur

rent

(mA

)

V15

200

600





GS-2013-Y (Physics)

Section C

To be attempted only by candidates for Ph.D. programme.

(Candidate for Integrated Ph.D. programme will get no credit for attempting this section.)

Q41. The integral

xxdx cos22

evaluates to

(a) -1 (b) 0 (c) 1

(d) 1

Q42. If z = x + iy then function

axxiyyxayxyxyxf 12111, 22

where a is a real parameter, is analytic in the complex z plane if a =

(a) -1 (b) +1 (c) 0 (d) i

Q43. A particle of mass m moves in one dimension under the influence of a potential

energy

42

llxbxaxV

where a and b are positive constants and ℓ is a characteristic length. The frequency of

small oscillations about a point of stable equilibrium is

(a) mb

21 (b)

mba 21

(c) ma

b 12

(d) ma

1





Q44. If a central force acting on a particle of mass m is given by

2rkrF

where r is the distance of the particle from the origin and k is a positive constant, the

Hamiltonian for the system, in spherical polar coordinates, will have the form

(a) rkrrrm 222222 sin

21

(b) mkrppprmr r 2

21 2222

2

(c) rk

mrp

mrp

mpr 2

22

2

22

2csc

22

(d) rk

mp

mp

mpr

222

222

Q45. The magnetic vector potential rA corresponding to a uniform magnetic field B

is taken in the form

rBA

21

where r is the position vector. If the electric field has the time-dependent form

,0tierEE

where ω is a constant, the gauge choice corresponding to this potential is

a

(a) non-linear gauge (b) Lorenz gauge

(c) time-varying gauge (d) Coulomb gauge

Q46. A binary star is observed to consist of a blue star B (peak wavelength 400 nm) and a

red star R (peak wavelength 800 nm) orbiting each other. As observed from the Earth, B

and R appear equally bright. Assuming that the stars radiate as perfect blackbodies, it

follows that the ratio of volumes VB / VR of the two stars is

(a) 64 (b) 1/64 (c) 16 (d) 1/16





Q47. An system at temperature T has three energy states 0, ± ε. The entropy of the system

in the low temperature (T → 0) and high temperature (T → ∞) limits are, respectively,

(a) ST→0 = 0 and ST→∞ = kB ln 3

(b) ST→0 = ST→∞ = kB ln 3

(c) ST→0 = 0 and ST→∞ = kB exp(-3)

(d) ST→0 = 0 and ST→∞ = 3kB /2

Q48. The state of a quantum mechanical system, in a certain basis, is represented by

the column vector

2/1

02/1

The operator A corresponding to a dynamical variable A, is given, in the same basis, by

the matrix

211121111

A

If, now a measurement of the variable A is made on the system in the state , the

probability that the result will be +1 is

(a) 41 (b)

21 (c)

21 (d) 1

Q49. A spin ½ particle A decays to two other particles B and C. If B and C are of spin ½

and spin-1 respectively, then a complete list of the possible values of the orbital angular

momentum of the final state (i.e. B + C) is

(a) 0, 1 (b) 23,

21 (c) 0, 1, 2 (d) 0, ±1





Q50. When a pure element is vaporized and placed in a uniform magnetic field B0, it is

seen that a particular spectral line of wavelength λ, corresponding to a J = 1 → J = 0

transition, gets split into three components λ, λ ±∆λ. It follows that the Lande g-factor for

the transition J = 1 → J = 0 is given by

(a)

2

0

BhcgB

(b) 20

BhcgB

(c)

2

0BhcgB

(d) 20

B

hcgB

Q51. The rate of deposition of a dielectric thin film on a thick dielectric substrate was

monitored by the following experiment:

a laser beam of wavelength λ = 633 nm,

at near-normal incidence θi, was

reflected from the thin film (see figure),

and the reflection coefficient R was

measured. As the film thickness

increased R varied with time as shown

in the figure. The refractive index of

the film is 3.07 and is less than that of

the substrate. Using the graph, the

approximate thickness of the film at the

end of 25 seconds can be estimated to

be

(a) 0.017 µm (b) 0.51 µm (c) 0.26 µm (d) 2.2 µm

0

10

20

30

40

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Film Substance

Ref

lect

ion

Coe

ffic

ient

(%)

i





Q52. The negative image on the right represents a very small portion of the night sky at a

very high resolution. Notice the broken ring(s) around the

central bright object in the middle of the picture. These

are most likely to be due to

(a) gravitational lensing of a distant object by the central

massive object

(b) gas clouds forming the remnant of a supernova explosion

(c) debris from a smaller object torn apart by tidal forces

(d) ice collected on the lens used for taking the picture

Q53. The magnetic susceptibility χ of three sample A, B and C, is measured as a function

of their absolute temperature T, leading to the graphs shown below.

From these graphs, the magnetic nature of the samples can be inferred to be

(a) A: anti-ferromagnet B: diamagnet C: paramagnet

(b) A: diamagnet B: paramagnet C: anti-ferromagnet

(c) A: paramagnet B: anti-ferromagnet C: ferromagnet

(d) A: anti-ferromagnet B: paramagnet C: ferromagnet

1

A

B

C

0 T





Q54. A gold foil, having N(0) number of 197Au nuclides per cm2, is irradiated by a beam

of thermal neutrons with a flux of F neutrons-cm-2-s-1. As a result, the nuclide 198Au, with

a half life τ of several years, is produced by the reaction

197Au + n → 198Au + γ

which has a cross section of σ cm2. Assuming that the gold foil has 100% abundancy of 197Au nuclides, the maximum number of 198Au nuclides that can accumulate at any time

in the foil is proportional to

(a) 0NF (b) 01 N

F (c) 0N

F (d) 0FN

Q55. The process of electron capture

p + e- → n + e

takes place at the quark level through the Feynman diagram

(a) (b)

(c) (d)

e

Wu

d

e eW

u

d

e

eW

u

d

e

Z

u

ed

e

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

Website: www.physicsbyfiziks.com Email: [email protected] 1

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16

TIFR-2014

Section-A

To be attempted by ALL candidates.

Q1. The directed beam from a small but powerful searchlight placed on the ground tracks a

small plane flying horizontally at a fixed height h above the ground with a uniform

velocity v , as shown in the figure below.

If the searchlight starts rotating with an instantaneous angular velocity 0 at time

0t when the plane was directly overhead, then at a later time t , its instantaneous

angular velocity t is given by

(a) t00 exp (b) t0

0

tan1

(c) 220

0

1 t

(d) 22

00

0

211 tt

Q2. The Conservation Principles for energy, linear momentum and angular momentum arise

from the necessity that

(a) the laws of physics should not involve infinite quantities.

(b) internal forces on a body should cancel out, by Newton’s (third) law of action and

reaction

(c) physical measurements should be independent of the origin and orientation of the

coordinate system.

(d) the laws of physics should be independent of the state of rest or motion of the

observer.

h

v





Q3. A uniform ladder of length L2 and mass m leans against

a wall in a vertical plane at an angle to the horizontal.

The floor is rough, having a coefficient of static friction

. A person of mass M stands on the ladder at a distance

D from its base (see figure). If the wall is frictionless,

the maximum distance maxD up the ladder that the

person can reach before the ladder slips is

(a) tan12

MmL (b) L

Mm

Mm

tan12

(c) tanL (d) tan2MmL

Q4. The product MN of two Hermitian matrices M and N is anti-Hermitian. It follows that

(a) 0, NM (b) 0, NM

(c) NM † (d) 1† NM

Q5. A student is asked to find a series approximation for the function xf in the

domain 11 x , as indicated by the thick line in the figure below

The student represents the function by a sum of three terms

2

sin2

cos 210xaxaaxf

Which of the following would be the best choices for the coefficients 10 ,aa and 2a ?

L2

D

1 0 1x

32

1

xf





(a) 0,31,1 210 aaa (b) 0,

32,

32

210 aaa

(c) 32,0,

32

210 aaa (d) 1,0,31

210 aaa

Q6. The probability function for a variable x which assumes only positive values is

xxxf exp

where 0 . The ratio xx ˆ/ , where x is the most probable value and x is the mean

value of the variable x , is

(a) 2 (b)

11 (c)

1 (d) 1

Q7. A solid spherical conductor has a conical hole made at one end, ending in

a point B , and a small conical projection of the same shape and size at the

opposite side, ending in a point A . A cross-section through the centre of

the conductor is shown in the figure on the right. If, now, a positive

charge Q is transferred to the sphere, then

(a) the charge density at both A and B will be undefined.

(b) the charge density at A will be the same as the charge density at B .

(c) the charge density at A will be more than the charge density at B .

(d) the charge density at B will be more than the charge density at A .

Q8. Solving Poisson’s equation 002 / for the electrostatic potential x in a region

with a con charge density 0 , two students find different answers, viz.

0

20

1 21

xx

and 02

02 21

yx

The reason why these different solutions are both correct is because

A

B





(a) space is isotropic and hence x and y are physically equivalent

(b) we can add solutions of Laplace’s equation to both x1 and x2

(c) the electrostatic energy is infinite for a constant charge density

(d) the boundary conditions are different in the two cases

Q9. A short solenoid with n turns per unit

length has diameter D and

length 15/8DL , as shown in the figure,

and it carries a constant current I . The

magnetic field B at a point P on the axis

of the solenoid at a distance 3/2DH

from its near end (see figure) is

2/122/32 11use xxxdx

(a) nl0654 (b) nl013

4 (c) nl01524 (d) nl065

112

Q10. A particle moving in one dimension has the un-noramalised wave function

2

2

exp

xxx

where is a real constant. The expectation value of its momentum is p

(a)

2

2

expx (b)

22

2

(c) 1exp (d) zero

Q11. A particle of mass m and charge e is in the ground state of a one-dimensional harmonic

oscillator potential in the presence of a uniform external electric field E . The total

potential felt by the particle is

eExkxxV 2

21

If the electric field is suddenly switched off, then the particle will

D

DL158

DH32

P





(a) make a transition to any harmonic oscillator state with keEx / as origin without

emitting any photon.

(b) make a transition to any harmonic oscillator state with 0x as origin and absorb a

photon.

(c) settle into the harmonic oscillator ground state with 0x as origin alter absorbing a

photon.

(d) oscillate back and forth with initial amplitude keE / , emitting multiple photons as it

does so.

Q12. Consider the Hamiltonian

xfH

Here x is the position vector, f is a constant and zyx ,, , where zyx ,, are

the three Pauli matrices. The energy ejgenvalues are

(a) zyxf 22 (b) iyxf

(c) 222 zyxf (d) zyxf

Q13. One mole of an ideal gas undergoes the cycle ACBA shown in the pV diagram below.

One of the curved lines in the cycle represents an isothermal change at temperatureT ,

while the other represents an adiabatic change. The net heat gained by the gas in this

cycle is

p

1p A

2p

1V 2V

C B

3VV





(a) 1

2232 ln

VVRTVVp (b)

1

3232 ln

VVRTVVp

(c) 11

12232 VVRTVVp (d)

1

32211 ln

VV

RTVpVp

Q14. An ideal gas at a temperature T is enclosed in a rigid container whose walls are initially

at temperature T where TT 1 . The wails are covered on the outside with perfect thermal

insulation and the system is allowed to come to equilibrium. The pressure exerted by the

gas on the walls of the container

(a) remains constant throughout.

(b) is lower at the initial stage than at the final stage.

(c) is higher at the initial stage than at the final stage.

(d) is the same at the initial and final stages.

Q15. Consider the CO molecule as a system of two point particles which has both translational

and rotational degrees of freedom. Using classical statistical mechanics, the molar

specific heat VC of CO gas is given in terms of the Boltzmann constant Bk by

(a) Bk25 (b) Bk2 (c) Bk

23 (d) Bk

21

Q16. In the laboratory frame, two observers A and B are moving along the sides of an

equilateral triangle with equal speeds 2/c , as shown in the figure. The speed of B as

measured by A will be

(a) c23 (b) c

334 (c) c

713 (d) c

35

A

B2/c

2/c





Q17. Two telescopes X and Y have identical objective lenses, but the single-lens eyepiece of

X is converging whereas the single-lens eyepiece of Y is diverging. If the magnification

M of these two telescopes for objects at infinity is the same, the lengths XL and YL of

the two telescopes (length of a telescope is defined as the distance between the objective

lens and the eyepiece) must be in the ratio yX LL /

(a) 1212

MM (b)

112

MM (c)

11

MM (d)

11

MM

Q18. A lens can be constructed using a flat circular glass plate whose refractive index n varies

radially, i.e. rnn , where r is the radial distance from the centre of the plate. In order

to make a convex lens by this method rn should vary (in terms of positive constants

0n and ) as

(a) 2/0 rn (b) rn /0 (c) rn 0 (d) 20 rn

Q19. A solid sample has the property that, when cooled below a certain temperature, it expels

any small applied magnetic field from within the material. Which of the following best

describes this sample in the cooled state?

(a) Paramagnet (c) Ferromagnet (b) Diamagnet (d) Anti-ferromagnet

Q20. A beam of atoms moving in a certain direction can be slowed down if they absorb

photons from a laser beam moving in the opposite direction and subsequently

spontaneously emit photons isotropically. For a beam of Sodium atoms (mass number

23A ) with speed 1600 ms , if a laser beam of wavelength 589 nm is used, the number

of such absorption and emission cycles needed to bring a Sodium atom to rest would be

approximately

(a) 5103.1 (b) 4103.1 (c) 3101.2 (d) 4101.2





Q21. In a laboratory, the double-slit experiment is performed with free non-relativistic

electrons, each having energy E , emitted from a source S (see figure below). The screen

consists of a uniform sheet of charge-sensitive pixels of size r . If the slit-screen distance

is z and the spacing between slits is d , which of the following restrictions on the

electron energy E should be satisfied so that the fringes can be distinctly observed?

(a) 2

21

rdhz

mE

e

(b) 2

21

rdhz

mE

e

(c)

rdhzcE (d)

rdhzcE

Q22. An alpha particle of energy E is shot towards a gold nucleus Au19779 . At distances much

larger than the nuclear size NR , the dominant force is the Coulomb repulsion, but at

distances comparable to the nuclear size the dominant force is the strong nuclear

attraction. These combine to form a potential barrier of height CV . If CVE , the

probability that the alpha particle will fuse with the gold nucleus can be written (in terms

of a dimensionless positive constant k ) as

(a) zero (b) 222

CVEkkE

(c)

CVEk 1 (d)

E

kVCexp

S dz sc

reen





Q23. Consider the following circuit.

Which of the graphs given below is a correct representation of outV ?

(a) (b)

(c) (d)

Q24. In the following circuit, the resistance 2R is doubled.

It follows that the current through 2R

(a) remains the same.

(b) is halved.

(c) is doubled

(d) is quadrupled.

tV 2sin10 outV

V5k1

0 1 2 3 4 510505

10

)( time t

Vou

t

0 1 2 3 4 510505

10

)( time t

Vou

t

0 1 2 3 4 510505

10

)( time t

Vou

t

0 1 2 3 4 510505

10

)( time t

Vou

t

inV

1R 2R





Q25. A control circuit needs to be designed to save on power consumption by an air-

conditioning unit A in a windowless room with a single door. The room is fitted with the

following devices:

1. a temperature sensor T , which is enabled 1T whenever the temperature falls

below a pre-set value;

2. a humidity sensor H which is enabled 1H whenever the humidity falls below a

certain pre-set value;

3. a sensor D on the door, which is triggered 1D whenever the door opens.

Which of the following logical circuits will turn the air-conditioning unit off 0A

whenever the door is opened or when both temperature and humidity are below their pre-

set values?

(a) (b)

(c) (d)

D

T

H

A

D

ATH

D

ATH

D

T

H

A





Section-B

To be attempted only by candidates for Integrated Ph.D. programme.

Q26. A body of mass m falls from rest at a height h under gravity (acceleration due to

gravity g ) through a dense medium which provides a resistive force 2kvF , where k

is a constant and v is the speed. It will hit the ground with a kinetic energy

(a)

mkh

kgm 2exp

2

2

(b) mkh

kgm 2tanh

2

2

(c)

mkh

kgm 2exp1

2

2

(d)

mkh

kgm 2exp1

2

2

Q27. A weight W is suspended from a rigid support by a hard spring

with stiffness constant K . The spring is enclosed in a- hard plastic

sleeve, which prevents horizontal motion, but allows vertical

oscillations (see figure). A simple pendulum of length with a

bob of mass Wmgm is suspended from the weight W and is

set oscillating in a horizontal line with a small amplitude. After

some time has passed, the weight W is observed to be oscillating

up and down with a large amplitude (but not hitting the sleeve). It

follows that the stiffness constant K must be

(a) WK 4

(b) WK 2

(c)

WK (d) 2

WK

Q28. In spherical polar coordinates ,,rr the delta function 21 rr

can be written as

(a) 212121 rr

(b) 21212121

coscos1 rr

r

l

m

KW





(c) 212121221

coscos1

rr

rr

(d) 2121211

21 cos

1

rr

r

Q29. An electric dipole is constructed by fixing two circular charged rings,

each of radius a, with an insulating contact (see figure). One of these

rings has total charge Q and the other has total charge Q . If the

charge is distributed uniformly along each ring, the dipole moment about

the point of contact will be

(a) zQa ˆ

(b) zQaˆ4

(c) zQaˆ2 (d) zero

Q30. A spherical conductor, carrying a total charge Q , spins uniformly and very rapidly about

an axis coinciding with one of its diameters. In the diagrams given below, the equilibrium

charge density on its surface is represented by the thickness of the shaded region. Which

of these diagrams is correct?

(a) (b)

(c) (d)

z

Q

Q





Q31. A rigid rotator is in a quantum state described by the wavefunction

sinsin43,

where and are the usual polar angles. If two successive measurements of zL are

made on this rotator, the probability that the second measurement will yield the value h

is

(a) 0.25 (b) 0.33 (c) 0.5 (d) negligible

Q32. A particle in the s2 state of hydrogen has the wave function

00

2/3

02 2

exp2124

1ar

ar

ars

where r is the radial coordinate w.r.t the nucleus as origin and 0a is the Bohr radius. The

probability P of finding the electron somewhere inside a sphere of radius 0a centered

at the nucleus, is best described by the graph

(a) (b)

(c) (d)

P

P

P

P





Q33. A thermally-insulated container of volume 0V is divided into two equal halves by a non-

permeable partition. A real gas with equation of state

nRTVapb

3

23

where a and b are constants, is confined to one of these halves at a temperature 0T . The

partition is now removed suddenly and the gas is allowed to expand to fill the entire

container. The final temperature of the gas, in terms of its specific heat VC will be

(a) 20

2

0 23

VCaTV

(b) 20

2

0 32

VCaTV

(c) 20

2

0 23

VCaTV

(d) 20

2

0 32

VCaTV

Q34. A manufacturer is able to offer two models of heat-conserving windows, as described

below.

Window A is a simple pane of glass, 4mm thick. Window B , on the other hand, consists

of two extremely thin panes of glass, separated by an air gap of 2mm, as shown in the

figure above. If the thermal conductivity of glass is known to be 118.0 KWm and that of

air is 11025.0 KWm , then the ratio of heat flow AQ through Window A to the heat flow

BQ through Window B is given by B

A

QQ

(a) 161 (b)

41 (c) 4 (d) 16

GlassIndoorOutdoor

4mmA Window

AirIndoor Outdoor

Glass Glass

2mmB Window





Q35. In a low temperature experiment, the resistance of a sensor is used as a thermometer. In

order to have better sensitivity in the range mK100 to K0.1 , which material would make

the best sensor?

(a) insulator (b) np junction

(c) pure semiconductor (d) metal

Q36. For a pure germanium semiconductor cooled in liquid nitrogen, the average density of

conduction electrons is about 31210 cmpern At this temperature the electron and hole

mobilities are equal and have the common value 1123100.5 sVcm ’. If a potential of

V100 is applied across a cm1 cube of this cooled germanium sample, the current

observed can be estimated as

(a) A80 (b) mA160 (c) mA16 (d) A16

Q37. A glass plate P (refractive index 54.1pn ) is coated with a dielectric material C with

the refractive index .6.1cn In order to have enhanced reflection from this coated glass

for near-normal incident light of wavelength , the thickness of the coating material C

must be

(a) even multiples of cn2

(b) even multiples of cn4

(c) odd multiples of cn4

(d) integral multiples of cn4

Q38. In the following circuit, the AC source is an ideal voltage source. What is the amplitude

of the steady state current through the inductor at resonance?

(a) LCV /0 (b) R

V0

(c) LCRCV 2/ 20 (d) zero

R

LC tVV sin0





Q39. A standard radioactive source is known to decay by emission of rays. The source is

provided to a student in a thick sealed capsule of unbreakable plastic and she is asked to

find out the half-life. Which of the following would be the most useful advice to the

student?

(a) The half-life cannot be measured because the initial concentration of the source is not

given.

(b) Mount the source in front of a gamma ray detector and count the number of photons

detected in one hour.

(c) Measure the mass of the source at different times with an accurate balance having a

least count of mg1 . Plot these values on a curve and fit it with an exponential decay law

(d) Mount the source in front of a gamma ray detector and count the number of photons

detected in a specific time interval. Repeat this experiment at different times and note

how the count changes.

Q40. Cosmic ray muons generated at the top of the Earth’s atmosphere decay according to the

radioactive decay law

2/1

693.0exp0T

tNtN

where tN is the number of muons at time t , and sT 52.12/1 is the proper half-life of

the muon. Immediately after generation, most of these muons shoot down towards the

Earth’s surface. Some of these muons decay on the way, but their interaction with the

atmosphere is negligible.

An observer on the top of a mountain of height km0.2 above mean sea level detects

muons with the speed c98.0 over a period of time and counts 1000 muons. The number

of muons of the same speed detected by an observer at mean sea level in the same period

of time would be

(a) 232 (b)539 (c) 839 (d) 983





Section-C

To be attempted only by candidates for Ph.D. programme.

Q41. The integral

044 x

dx

evaluates to

(a) (b)2 (c)

4 (d)

8

Q42. The solution of the integral equation

x

tfdtxxf0

has the graphical form

(a) (b)

(c) (d)

0

2.0

4.0

6.0

8.0

1

0 1 2 3 4 5

xf

x0

2.0

4.0

6.0

8.0

1

0 1 2 3 4 5

xf

x

0

2.0

4.0

6.0

8.0

1

0 1 2 3 4 5

xf

x

xf

x0

2.0

4.0

6.0

8.0

1

0 1 2 3 4 5





Q43. Assume that the Earth is a uniform sphere of radius R , rotating about its axis with a

uniform angular velocity . A rocket is launched from the Equator in a direction due

North. If it keeps on flying at a uniform speed v (neglecting air resistance), the highest

latitude that can be achieved is

(a) 2 (b)

vR

22

(c) vR 2

2 (d)

vR 21

2

Q44. A particle p of mass m moves under the influence of a central potential, centred at the

origin O , of the form 33rkrV where k is a positive constant

If the particle P comes in from infinity with initial velocityu and impact parameter b

(see figure), then the largest value of b for which the particle gets captured by the

potential is

(a) 6/1

42

23

um

k (b) 3/1

3

muk (c)

6/1

42

22

um

k (d) 3/1

32

muk

Q45. The instantaneous electric and magnetic fields created at a distance r by a point source at

the origin are given by

ˆcos,ˆ

2cos

00 rtBH

rtAE

where BA,, are constants, and the unit vectors ˆ,ˆ,r form an orthonormal set. The

time-averaged power radiated by the source is

(a) AB0

0

(b) ABc

2

3

(c) ABc 2 (d) ABc2

O

u p

b





Q46. A particle is confined to a one-dimensional box of length L . If a vanishingly thin but

strongly repulsive partition is introduced in the exact centre of the box, and the particle is

allowed to come to its ground state, then the probability density for finding the particle

will appear as

(a) (b)

(c) (d)

Q47. A spin-2 nucleus absorbs a spin-21 electron and is then observed to decay to a stable

nucleus in two stages, recoiling against an emitted invisible particle in the first stage and

against an emitted spin-1 photons in the second stage. If the stable nucleus is spinless,

then the set of all possible spin values of the invisible particle is

(a)

25,

21 (b)

27,

23

(c)

25,

23,

21 (d)

27,

25,

23,

21

2x

x

2x

x

2x

x

2x

x





Q48. A gas of photons is enclosed in a container of fixed volume at an absolute temperatureT .

Noting that the photon is a massless particle (i.e., its energy and momentum are related

by pcE ), the number of photons in the container will vary as

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

Q49. In a Stern-Gerlach experiment with spin-21 particles, the beam is found to form two spots

on the screen, one directly above the other. The experimenter now makes a hole in the

screen at the position of the upper spot. The particles that go through this hole are then

passed through another Stern-Gerlach apparatus but with its magnets rotated by 90

degrees counter clockwise about the axis of the beam direction. Which of the following

shows what happens on the second screen?

(a) (b)

(c) (d)

Q50. The ground state electronic configuration for a carbon atom is

. 222 221 pss

The first excited state of this atom would be achieved by

(a) re-alignment of the electron spins within the p2 orbital

(b) transition of an electron from the s2 orbital to the p2 orbital

(c) transition of an electron from the p2 orbital to the s3 orbital.

(d) transition of an electron from the s2 orbital to the s3 orbital





Q51. Which of the following statement best explains why the specific heat of electrons in

metals is much smaller than that expected in a non-interacting (free) electron gas model?

(a) The mass of electron is much smaller than that of the ions in the crystal.

(b) The Pauli Exclusion Principle restricts the number of electrons which can absorb

thermal energy.

(c) Electron spin can take only two different values.

(d) Electrons in a metal cannot be modelled as non-interacting.

Q52. In a beta decay experiment, an electromagnet M and a detector D are used to measure

the energy of electrons , as shown in the figure.

The detector D is capable of detecting either electrons or positrons . Now the source is replaced with a source, and we would like to measure the energy of the

positrons using the same setup. Which of the following is correct?

(a) This can be done quite easily, if-the polarity of current in the coils of the

electromagnet is reversed.

(b) This can be done trivially, without changing anything, since the detector D can detect

either or .

(c) There is no way to do this with the given set up, since will have to be converted

into , which is obviously not possible.

(d) This cannot be done since the magnet does not have a symmetric shape.

D

M

S





Q53. It is well-known that the energy of the Sun arises from the fusion of hydrogen nuclei

(protons) inside the core of the Sun. This takes place through several mechanisms, each

resulting in emission of energy.

Which of the following reactions is NOT possible during the proton fusion inside the

Sun?

(a) HeHH 22

11

11 (b) HeHH 3

211

21

(c) eeHHH 21

11

11 (d) eHeHHHH 24

211

11

11

11

Q54. A group of alien astronomers far away from the solar system tries to find out in the Sun

(visible to them as a small yellow star) has planets orbiting around it. The method they

use is to look for wobbles in the motion of the Sun induced by the planet(s) revolving

around it (if any). To detect this motion, they build a high-resolution spectrometer which

can measure the Doppler shift in frequency of a 600 nm line in the solar spectrum with an

accuracy of 1 in 610 .

Given that the Sun has a mass kg30102 arid that Earth (mass kg24106 , orbital

velocity 14103 ms and Jupiter (mass kg27102 , orbital velocity 14105.1 ms ) are two

typical planets, one could predict that the experiments conducted by the aliens would find

(a) evidence for both the planets Earth and Jupiter.

(b) evidence for the planet Jupiter, but not for the planet Earth.

(c) no evidence for any planets orbiting the Sun.

(d) evidence for planets, but will not be able to tell how many.

Q55. The interaction strength of the recently-discovered Higgs boson (mass

approximately 2/125 cGeV ) with any other elementary particle is proportional to the

mass of that particle. Which of the following decay processes will have the greatest

probability?

(a) Higgs boson decaying to a top quark a top anti-quark.

(b) Higgs boson decaying to a bottom quark a bottom anti-quark.

(c) Higgs boson decaying to an electron and a positron.

(d) Higgs boson decaying to a neutrino-antineutrino pair.





TIFR - 2015

Q1. Which of the following vectors is parallel to the surface 422 xzyx at the point

3,2,2 ?

(a) kji ˆ5ˆ2ˆ6 (b) kji ˆ5ˆ2ˆ6

(c) kji ˆ5ˆ2ˆ6 (d) kji ˆ5ˆ2ˆ6

Q2. A random number generator outputs 1 or 1 with equal probability every time it is

run. After it is run 6 times, what is the probability that the sum of the answers generated

is zero? Assume that the individual runs are independent of each other.

(a) 32

15 (b)

16

5 (c)

6

5 (d)

2

1

Q3. It is required to construct the quantum theory of a particle of mass m moving in one

dimension x under the influence of a constant force F . The characteristic length-scale in

this problem is

(a) 2

mF (b)

mF

(c)

3/1

2

Fm

(d)

3/12

mF

Q4. A particle slides on the inside surface of a frictionless cone. The cone is fixed with its tip

on the ground and its axis vertical, as shown in the figure on the right. The semi-vertex

angle of the cone is . If the particle moves in a circle of radius 0r , without slipping

downwards, the angular frequency of this motion will be

(a) tan0r

g (b)

cot0r

g

(c) sin0r

g (d)

ar

g

cos0

0r





Q5. A car starts from rest and accelerates under a force F increasing linearly in time as

atF at where a is a constant. At time ,01 t , the force F is suddenly switched off.

At a later time 12 tt , brakes are applied resulting in a force F whose magnitude

increases linearly with time, 2ttaF where a is the same constant as before.

Which of the following graphs would best represent the change in the position of the car

tx with time?

(a) (b)

(c) (d)

Q6. In the Earth’s atmosphere, a localised low-pressure region (shaded in diagrams) develops

somewhere in the southern hemisphere. Which one of the following diagrams represents

the correct air flow pattern as observed from a satellite?

(a) (b)

(c) (d)

0 1t 2t 12 tt

tx

t0 1t 2t 12 tt

tx

t

0 1t 2t 12 tt

tx

t0 1t 2t 12 tt

tx

t





Q7. The focal length in air of a thin lens made of glass of refractive index 5.1 is . When

immersed in water (refractive index3

4 ), its focal length becomes

(a) 4

3 (b)

3

4 (c)

4

(d) 4

Q8. A light beam of intensity 0I passes at normal incidence through a flat plate of plastic

kept in air. If reflection at the interface reduces the intensity by %20 and absorption on

passing through the plate reduces the intensity by %2 , the intensity of the emergent

beam will be about

(a) 078.0 I (b) 00.65 I (c) 00.63 I (d) 00.60 I

Q9. A light beam is propagating through a medium with index of refraction 5.1 . If the

medium is moving at constant velocity c7.0 in the same direction as the beam, what is

the velocity of light in the medium as measured by an observer in the laboratory? ( c

velocity of light in vacuum)

(a) c96.0 (b) c98.0 (c) c93.0 (d) c90.0

Q10. Two blackbodies radiate energy at temperatures 1T and 2T 21 TT . The energy emitted

per unit time per unit solid angle per unit surface area of a blackbody in the frequency

range v to dvv is given by dvvB . Which one of the following graphs has the correct

form?

(a) (b) 1T

2T

(c) (d)

v0

vB

1T

2T

v0

vB

1T

2T

v0

vB

1T

2T

v0

vB





Q11. In a cold country, in winter, a lake was freezing slowly. It was observed that it took 2

hours to form a layer of ice 2 cm thick on the water surface. Assuming a constant

thermal conductivity throughout the layer, the thickness of ice would get doubled after

(a) 8 more hours. (b) 4 more hours.

(c) 2 more hours. (d) 6 more hours.

Q12. Which of the following graphs qualitatively describes the pressure P of a gas of non-

interacting fermions in thermal equilibrium at a constant volume as a function of

temperature?

(a) (b)

(c) (d)

Q13. The electrostatic potential r of a distribution of point charges has the form 3 rr

at a distance r from the origin 0,0,0 , where ar . Which of the following

distributions can give rise to this potential?

(a) (b)

(c) (d)

P

T

P

TP

T

P

T

q q

q qa

0,0,0

q

q

q

qa

0,0,0

q q

qq

a

0,0,0

q

qq

q

a

0,0,0





Q14. Two semi-infinite solenoids placed next to each other are separated by a small gap of

width W as shown in the figure.

The current 0I in the solenoids flows in the direction as shown. If the solenoids have a

circular cross-section of radius R and are filled with a magnetic material of permeability

0 then the magnetic energy densities iu inside the solenoid and gu in the gap

are best related by

(a) ig cuu (b) ig cuu (c) ig uu (d) ig uu

Q15. A light source has a small filament at the centre of a spherical glass bulb of radius cm5

and negligible thickness. If this source emits 100 Watts of power in the form of spherical

electromagnetic waves, the smr .. electric field E at the surface of the bulb (in units of

Volt/ m ) will be approximately

(a) 1094 (b) 4.109 (c) 47.15 (d) 1547

Q16. A particle is moving in one dimension under a potential xV such that, for large positive

values of kxxVx , where 0k and 1 . If the wavefunction in this region has the

form xx exp~ which of the following is true?

(a) (b) 12

(c) 22 (d)

2

2

Q17, The ground state energy of a particle of mass m in a three-dimensional cubical box of

side is not zero but 2

2

8

3

m

hThis is because

(a) the potential at the boundaries is not really infinite, but just very large.

(b) this is the most convenient choice of the zero level of potential energy.

(c) position and momentum cannot be exactly determined simultaneously.

(d) the ground state has no nodes in the interior of the box.

W

0I0I





Q18. A one-dimensional box contains a particle whose ground state energy is . It is observed

that a small disturbance causes the particle to emit a photon of energy 8hv , after

which it is stable. Just before emission, a possible state of the particle in terms of the

energy eigenstates 1 2, ,... would be

(a) 1 2 52 3 5

6

(b) 2 32

5

(c) 4 54 5

41

(d) 1 2

2

l

Q19. A sample of ordinary hydrogen H1

1 gas in a discharge tube was seen to emit the usual

Balmer spectrum. On careful examination, however, it was found that the aH line in the

spectrum was split into two fine lines, one an intense line at nm28.656 , and the other a

faint line at 656.04 nm . From this, one can conclude that the gas sample had a small

impurity of

(a) H3

1 (b) H2

1 (c) He4

2 (d) OH 2

Q20. An electron makes a transition from the valence band to the conduction band in an

indirect band gap semiconductor. Which of the following is NOT true?

(a) There is no momentum change in the electron.

(b) A phonon is involved in the process.

(c) A photon is absorbed in the process.

(d) The energy of the electron increases.

Q21. Which of the following radioactive decay chains is it possible to observe?

(a) AuHgPb 202

79

202

80

206

82 (b) PbBiPb 206

82

206

83

206

82

(c) PbPoBi 206

82

210

84

210

83 (d) 214 210 207

88 86 82Ra Rn Pb

Q22. Which of the following is the best technique for measuring the effective mass of an

electron in a semiconductor?

(a) Millikan’s oil drop experiment (b) Cyclotron resonance

(c) X -ray diffraction experiment (d) Resistivity measurements





Q23. Two LCR circuits A and B are shown below where CCc . At time 0t , a

charge Q is put on the capacitor C .

Which of the following statements is correct?

(a) The charge Q will decay at the same rate in A and B

(b) The charge Q will decay faster in B

(c) The charge Q will decay faster in A

(d) The relative decay rates cannot be predicted without knowing the exact values of

RCL ,, and cC

Q24. In the circuit shown below, the op-amp is powered by a bipolar supply of V10 .

Which one of the following graphs represents outV correctly?

(a) (b)

(c) (d)

tV 2sin5

k5

outV

outV 0

5

10

10

5

0 1 2 3 4 5t

outV 0

5

10

10

5

0 1 2 3 4 5t

outV 0

5

10

10

5

0 1 2 3 4 5t

outV 0

5

10

10

5

0 1 2 3 4 5t

LC

cC

R

)()A(

LCR





Q25. All resistors in the circuit on the right have a

tolerance of %5 .

Assuming a diode drop of 0.7V , which of the following is

the lowest possible value of the collector voltage?

(a) 4.7V (b) 5.2V

(c) 3.1V (d) 4.1V

GS-2015(Physics)

Section B

Q26. Consider the differential equation

dx

dyy

dx

yd4

2

2

with the boundary condition that 0xy at 5

1x . When plotted as a function of x , for

0x , we can say with certainty that the value of y

(a) first increases, then decreases to zero

(b) first decreases, then increases to zero

(c) has an extremum in the range 10 x

(d) oscillates from positive to negative with amplitude decreasing to zero

Q27. In a transmission diffraction grating, there are 410 lines/mm. Which of the following

ranges of wavelength (in nm ) will produce at least one principal maximum?

(a) 50001001 (b) 1000501 (c) 500201 (d) 2001

V10

k150

k51k100

k330





Q28. Two cylinders A and B of the same length L and outer

radius R were placed at the same height h on an

inclined plane at an angle with the horizontal (see

figure). Starting from rest, each cylinder was allowed to

roll down the plane without slipping. It was found that

A reached the end of the inclined plane earlier than B

Which of the following possibilities could be true?

(a) A is solid and made of copper; B is hollow and made of copper.

(b) A is solid and made of copper; B is solid and made of aluminium.

(c) A is hollow and made of aluminium; B is solid and made of aluminium.

(d) A is hollow and made of copper; B is hollow and made of copper; B is heavier

than A .

Q29. A thin uniform rod of length l2 and mass M is pivoted at one end P on a horizontal

plane (see figure). A ball of mass Mm and speed 0v strikes the free end of the rod

perpendicularly and bounces back with velocity fv along the original line of motion as

shown in the figure. If the collision is perfectly elastic the magnitude of fv is

(a) 03

3v

mM

mM

(b) 03

3v

mM

mM

(c) 04

4v

mM

mM

(d) 04

4v

mM

mM

L

h

0VfV

M 2t

P





Q30. A collimated beam of pions originate from an accelerator and propagates in vacuum

along a long straight beam pipe. The intensity of this beam was measured in the

laboratory after a distance of 75m and found to have dropped to one-fourth of its

intensity at the point of origin. If the proper half- life of a pion is s81077.1 , the speed

of the pions in the beam, as measured in the laboratory, must be

(a) c99.0 (b) c98.0 (c) c96.0 (d) c97.0

Q31. The equation of state of a gas is given by

T

b

P

RTV

where R is the gas constant and b is another constant parameter. The specific heat at

constant pressure PC and the specific heat at constant volume VC for this gas is related

by VP CC

(a) R (b)

22

1

bP

RTR

(c)

2

21

RT

bPR (d)

2

21

RT

bPR

Q32. An ideal diatomic gas is initially at a temperature CT o0 . Then it expands reversibly

and adiabatically to 5 times its volume. Its final temperature will be approximately

(a) Co0 (b) C0150 (c) Co130 (d) Co180

Q33. Consider an infinitely long cylinder of radius R , placed along the z -axis, which carries a

static charge density krr , where r is the perpendicular distance from the axis of

the cylinder and k is a constant. The electrostatic potential r inside the cylinder is

proportional to

(a)

1

3

23

3

R

r (b) 2 ln

r

R

(c)

1

3

23

3

R

r (d) 2 ln

R

r





Q34. A solid spherical conductor encloses 3 cavities, a cross-section

of which are as shown in the figure. A net charge q resides on

the outer surface of the conductor. Cavities A and C contain

point charges q and q , respectively.

The net charges on the surfaces of these cavities are

(a) qCBqA ,0, (b) 0,, CqBqA

(c) qCBqA ,0, (d) qCBqA ,0,

Q35. 1000 neutral spinless particles are confined in a one-dimensional box of length nm100 .

At a given instant of time, if 100 of these particle have energy 04 and the remaining

900 have energy 0225 then the number of particles in the left half of the box will be

approximately

(a) 441 (b) 100 (c) 500 (d) 625

Q36. A one-dimensional quantum harmonic oscillator is in its ground state

2

4/1

0

2xm

em

x

Two experiments, A and B are performed on the system. In A the frequency of

the oscillator is suddenly doubled, while in B the frequency is suddenly halved. If

Ap and Bp denote the probability in each case that the system is found in its new ground

state immediately after the frequency change, which of the following is true?

(a) BA pp 2 (b) BA pp (c) BA pp 2 (d) BA pp 2

Q37. In the basic band structure theory of crystalline solids, which of the following leads to

energy gaps in the allowed electronic energy values?

(a) Bragg reflection (b) Electron spin

(c) Electron-phonon interaction (d) Electron-electron interaction

qB

CAq

q





Q38. The material inside a box is either a metal or a semiconductor. If 1R is the

resistance of the material, which of the following experiments CANNOT distinguish

whether it is a metal or a semiconductor?

(a) Measurement of R at different temperatures.

(b) Measurement of R using power supplies of different frequencies.

(c) Measurement of absorption spectrum in the energy range eV21.0 .

(d) Measurement of R in the presence of different magnetic fields.

Q39. To measure the voltage in the range 0 5V with a precision of 5 mV , the minimum

number of bits required in a digital voltmeter is

(a) 12 (b) 9 (c) 11 (d) 10

Q40. A building has three overhead water tanks, each fitted with a sensor ( 321 ,, SSS ) which

goes to 0 when the water level in the tank falls below a set value and remains 1

otherwise. A common pump is used to raise water from an underground storage tank to

these overhead tanks. Of the following circuits, which one will turn on 1P the pump

only when at least two of the tanks have water level below the set value?

(a) (b)

(c) (d)

P3S

2S1S

P3S

2S1S

P

3S2S

1S

3S2S

1S

P3S

2S1S





GS-2015-Y Physics

Section C

Q41. The integral

2

0 2cos21 aa

d where 10 a , evaluate to

(a) 21

2

a

(b)

21

2

a

(c) 2 (d)

21

4

a

Q42. The generating function for a set of polynomials in x is given by

1221,

txttxf

The third polynomial (order 2x ) in this set is

(a) 12 2 x (b) 14 2 x (c) xx 22 (d) 14 2 x

Q43. A particle moves under the influence of a central potential in an orbit 4kr ,where k

is a constant and r is the distance from the origin. It follows that the angle varies with

time t as

(a) 7

1

t (b) 8

1

t (c) 9

1

t (d) 6

1

t

Q44. In a system with two degrees of freedom, if qp, are the canonical coordinates, then

which of the following transformations to QP, is canonical?

(a) p

qQqpP

2tan,

2

1 122 (b) q

pQqpP 122 cot,

2

1

(c) p

qQqpP

2sin,

2

1 122 (d) q

pQqpP 122 cos,

2

1

Q45. In a monatornic gas, the first excited state is only 1.5eV above the ground state, while

the other excited states are much higher up. The ground state is doubly-degenerate, while

the first excited state has a four-fold degeneracy. If now, the gas is heated to a

temperature of 7000 K , the fraction of atoms in the excited state will be approximately

(a) 42.0 (b) 3.0 (c) 14.0 (d) 07.0





Q46. Measurement of the magnitudes of the electric field E and the magnetic field B in a

plane-polarised electromagnetic wave in vacuum leads to the following results

t

B

y

E

t

E

cy

B

2

1

at all points where the measurement is made. In this case the electric vector E

, the

magnetic vector B

and the wave vector k

(with magnitude k ) can be written in terms of

the unit vectors ( zyx ˆ,ˆ,ˆ ) along the Cartesian axes as

(a) ykkzBBxEE ˆ,ˆ,ˆ

(b) ykkzBBxEE ˆ,ˆ,ˆ

(c) zkkyBBxEE ˆ,ˆ,ˆ

(d) xkkzBByEE ˆ,ˆ,ˆ

Q47. A two-state quantum system has two observables A and B . It is known that the

observable A has eigenstates 1 and

2 with eigenvalues 1a and 2a respectively,

while B has eigenstates 1 and

2 with eigenvalues 1b and 2b respectively, and that

these eigenstates are related by

2115

4

5

3 212

5

3

5

4

Suppose a measurement is made of the observable A and a value 1a is obtained. If the

observable B is now measured, the probability of obtaining the value 1b will be

(a) 80.0 (b) 64.0 (c) 0.60 (d) 36.0

Q48. An rigid rotator has the wave function

,3,2,2, 1,11,20,1 iYYiiYN

where ,, mlY are the spherical harmonics, and N is a normalization constant. If L

is

the orbital angular momentum operator, and yx iLLL the expectation value of LL

is

(a) 9

21 2 (b)

9

23 2 (c)

9

25 2 (d) 0





Q49. In the ground state electronic configuration of nitrogen N14

7 the SL, and J quantum

numbers are

(a) 2

3,

2

1,1 JSL (b)

2

3,

2

3,0 JSL

(c) 2

1,

2

1,0 JSL (d)

2

1,

2

1,1 JSL

Q50. Solar radiation tends to push any particle inside solar system away from the Sun.

Consider a spherical dust particle of specific gravity 0.6 and no angular momentum

about the Sun. What should be its minimum radius so that it does not escape from the

solar system? Take the solar luminosity to be W26108.3 .

(a) 0.1 m (b) 10 m (c) 0.01 m (d) m610

Q51. Bosonic excitations of ferromagnets have a dispersion relation 2ke where is the

energy and k is the wavevector of the excitation. Assuming a system of such non-

interacting bosonic excitations, at low temperature T , the specific heat VC of a three-

dimensional ferromagnet will be proportional to

(a) T (b) 2

5

T (c) 3T (d) 2

3

T

Q52. In the semi-empirical mass formula, the volume V , surface S , coulomb C , and

pairing P contributions to the binding energy of a nucleus XA

Z vary with mass number

A as

(a) 4

3

3

1

3

2

,,,

APACASAV

(b) 4

3

3

1

3

1

,,,

APACASAV

(c) 4

3

3

1

3

2

,,,

APACASAV

(d) 4

3

3

1

3

2

2 ,,,

APACASAV





Q53. In an experiment, Au197

79 nuclei were bombarded with neutrons leading to formation of

Au198

79 , which is unstable. The half-life of Au198

79 was measured to be 25.2 days and it

was found later that this measured half-life was an underestimate by %10 . The

corresponding percentage error in the estimated population of Au198

79 after 9 day’s is

(a) %5.2 (b) %10 (c) %15 (d) %25

Q54. You are given the following circuit and two instruments: a voltmeter and an ammeter

both with %001.0 accuracy in their readings.

Which of the following methods will result in the most accurate reading for the current

without interrupting the current in the circuit?

(a) Use voltmeter to measure voltage across points A and B

(b) Use the ammeter to measure current at point B

(c) Use voltmeter to measure voltage across points B and C

(d) Use voltmeter to measure voltage across points A and C

Q55. Consider the following reactions involving elementary particles:

(A) Kp

(B) pKpK

Which of the following statements is true for strong interactions?

(a) A and B are both forbidden

(b) B is allowed but A is forbidden

(c) A is allowed but B is forbidden

(d) A and B are both allowed

A B C

5V

10 0.001 100 1

1

GS-2016 (Physics)TATA INSTITUTE OF FUNDAMENTAL RESEARCH

Written Test in PHYSICS December 13, 2015Duration : Three hours (3 hours)

Name : _______________________________________________ Ref. Code : ____________


1. Please fill in details about name, reference code etc. on the question paper and answer sheet. The Answer Sheet is machine-readable. Use only black/blue ball point pen to fill in the answer sheet.

2. This test consists of three parts, Section A, Section B and Section C. You must answer questions according to the program you are applying for :

Candidates applying for Must Answer Should Not Attempt

Integrated Ph.D. Section A and Section B Section C

Ph.D. Section A and Section C Section B

Section A contains 25 questions, Section B and Section C contain 15 questions each. Note that the test contains multiple-choice questions as well as numerical-type questions (36-40 in section B and 51-55 in section C) where you have to fill in numbers in the answer-sheet.

4. Indicate your ANSWER ON THE ANSWER SHEET by filling in the appropriate circle or circles completely for each question.

For multiple-choice questions, only ONE of the options given at the end of each question is correct. Do not mark more than one circle for any multiple choice question : this will be treated as a wrong answer.

For number-type questions, the answers should be indicated on the answer sheet by filling in circles for appropriate numbers. All three circles for each answer should be filled to get the credit. Detailed instructions are given inside the question paper.

5 . The marking shall be as follows:

Multiple-choice Numerical

If the answer is correct : +3 +5

If the answer is incorrect : -1 0

If the answer is not attempted : 0 0

If more than one circle is marked (only for multiple-choice): 0 Not applicable

Note that there is negative marking for multiple-choice questions, but not for the numerical-type questions.

6. We advise you to first mark the correct answers on the QUESTION PAPER and then to TRANSFER these to the ANSWER SHEET only when you are sure of your choice.

7. Rough work may be done on blank pages of the question paper. If needed, you may ask for extra rough sheets from an Invigilator.

8. Use of calculators is permitted. Calculator which plots graphs is NOT allowed. Multiple-use devices such as cell phones, smartphones etc., CANNOT be used for this purpose.

9. Do NOT ask for clarifications from the invigilators regarding the questions. They have been instructed notto respond to any such inquiries from candidates. In case a correction/clarification is deemed necessary, the invigilator(s) will announce it publicly.

10. List of useful physical constants is given on the next page. Make sure to use only these values while answering the questions.

PHY - XCorrect answers are ticked in green.

2

USEFUL CONSTANTS

UNIT CONVERSIONS

Symbol Name/Definition Value

speed of light in vacuum m s-1 reduced Planck constant ( ) J s gravitational constant m3 kg-1 s-2 solar mass kg

permittivity of free space F m-1 permeability of free space N A-2 electron charge (magnitude) C electron mass kg

MeV/c 2 Bohr radius 0.510 Å

ionisation potential of H atom 13.6 eV Avogadro number mol-1 Boltzmann constant J K-1

eV K-1 gas constant J mol-1 K-1 ratio of specific heats: monatomic gas 1.67 diatomic gas 1.40

Stefan-Boltzmann constant W m-2 K-4 fine structure constant ( ) 1/137 acceleration due to gravity 9.80 m s-2 radius of the Earth Km radius of the Sun Km proton mass ( ) kg

MeV/c 2 neutron mass ( ) kg

MeV/c 2

Symbol Name/Definition Value

1 A.U. mean distance of Earth from Sun km 1 a.m.u. atomic mass unit kg

MeV/c 2

1 eV electron Volt J

1 T Tesla gauss

1 bar mean atmospheric pressure at 00 C Pa (= N m-2)

1 Å Ångstrom unit 10-8 cm

conversion constant J m-1

GeV fm

3

GS-2016-X (Physics) Section A

To be attempted by ALL candidates.

1. If is a continuous variable which is uniformly distributed over the real line from to according to the distribution

then the expectation value of

is

(a) zero (b) (c) (d)

2.

If the eigenvalues of a symmetric matrix A are and the corresponding eigenvectors can be written as

respectively, then the matrix is

(a)

(b)

(c)

(d)

3.

In a triangular lattice a particle moves from a lattice point to any of its 6 neighbouring points with equal probability, as shown in the figure on the right. The probability that the particle is back at its starting point after 3 moves is

(a) (b) (c) (d)

4.

A ball is dropped vertically from a height on to a plane surface and permitted to bounce repeatedly along a vertical line. After every bounce, its kinetic energy becomes a quarter of its kinetic energy before the bounce. The ball will come to rest after time

(a) infinity (b) (c) (d)

4

5.

In a moving car, the wheels will skid if the brakes are applied too suddenly. This is because

(a) the inertia of the car will carry it forward.

(b) the momentum of the car must be conserved.

(c) the impulsive retarding force exceeds the limiting force of static friction.

(d) the kinetic friction will suddenly get converted to static friction.

6. An aircraft, which weighs kg when unloaded, is on a relief mission, carrying food packets weighing 1 kg each. The plane is gliding horizontally with its engines off at a uniform speed of kmph when the first food packet is dropped. Assume that the horizontal air drag can be neglected and the aircraft keeps moving horizontally. If one food packet is dropped every second, then the distance between the last two packet drops will be

(a) Km (b) m (c) m (d) m

7.

Imagine that a narrow tunnel is excavated through the Earth as shown in the diagram on the left and that the mass excavated to create the tunnel is extremely small compared to Earth’s mass .

A person falls into the tunnel at one end. at time . Assuming that the tunnel is frictionless, the person will

(a) fall straight through, escaping Earth's gravity at time (b) describe simple harmonic motion with period (c) describe simple harmonic motion with period (d) describe simple harmonic motion with period 8.

A grounded conducting sphere of radius is placed with its centre at the origin. A point dipole of dipole moment is placed at a distance along the -axis, where are the units vector along the and -axes respectively. This leads to the formation of an image dipole of strength at a distance from the centre along the -axis. If , then

(a) (b) (c) (d)

5

9.

A long, solid dielectric cylinder of radius a is permanently polarized so that the polarization is everywhere radially outward, with a magnitude proportional

to the distance from the axis of the cylinder, i.e., . The bound

charge density in the cylinder is given by

(a) (b) (c) (d)

10. A circular loop of fine wire of radius carrying a current is placed in a

uniform magnetic field perpendicular to the plane of the loop. If the breaking tension of the wire is , the wire will break when the magnetic field exceeds

(a) (b)

(c) (c)

11. A gas of non-interacting particles, each of rest mass MeV, is at a

temperature K and has an average particle density cm-3. We can obtain a reasonably correct treatment of this system

(a) only by using special relativity as well as quantum mechanics.

(b) by neglecting quantum mechanics but not special theory of relativity.

(c) by neglecting special relativity but not quantum mechanics.

(d) by neglecting both special relativity and quantum mechanics.

12.

A one-dimensional harmonic oscillator of mass and natural frequency is in the quantum state

at time , where denotes the eigenstate with eigenvalue . It follows that the expectation value of the position operator is

(a) (b)

(c) (d)

6

13.

A particle is confined inside a one-dimensional potential well , as shown on the right. One of the possible probability distributions

for an energy eigenstate for this particle is

(a)

(b)

(c)

(d)

14.

The energy per oscillator of an isolated system of a large number of identical non-interacting fermions in a one-dimensional harmonic oscillator potential is

, where is the angular frequency of the harmonic oscillator. The entropy of the system per oscillator is given by

(a) (b) (c) (d)

15.

In the temperature range C, the molar specific heat of a metal varies with temperature (measured in degrees Celsius) according to the formula J-deg C-1-mol-1. If kg of the metal at C is brought in thermal contact with kg of the same metal at C, the final equilibrium temperature, in deg C, will be

[Assume that no heat is lost due to radiation and/or other effects.]

(a) (b) (c) (d)

16.

In a fixed target experiment, a proton of total energy 200 GeV is bombarded on a proton at rest and produces a nucleus and its anti-nucleus

The heaviest nucleus that can be created has atomic mass number

(a) (b) (c) (d)

7

17.

In a simple cubic lattice of lattice constant 0.287 nm, the number of atoms per mm2 along the 111 plane is

(a) (b) (c) (d)

18.

In a glass fibre, light propagates by total internal reflection from the inner surface. A very short pulse of light enters a perfectly uniform glass fibre at

and emerges from the other end of the fibre with negligible losses. If the refractive index of the glass used in the fibre is 1.5 and its length is exactly 1.0 km, the time at which the output pulse will have completely exited the fibre will be

(a) (b) (c) (d) 19. In an ionization experiment conducted in the laboratory, different singly-

charged positive ions are produced and accelerated simultaneously using a uniform electric field along the -axis. If we need to determine the masses of various ions produced, which of the following methods will NOT work

(a) Detect them at a fixed distance from the interaction point along -axis and measure their time of arrival.

(b) Apply a uniform magnetic field along -axis and measure the deviation.

(c) Apply a uniform electric field along -axis and measure the deviation.

(d) Apply a uniform electric field along -axis and a (variable) uniform magnetic field along -axis simultaneously and note the zero deviation.

20.

An observer stands at the edge of a swimming pool, as sketched in the figure below.

This observer will perceive the pool as

(a)

(b)

(c)

(d)

8

21.

Consider a process in which atoms of Actinium-226 ( ) get converted to atoms of Radium-226 ( ) and the yield of energy is MeV per atom. This occurs through

(a)

and (b)

and

(c) Only (d) Only

22. Two homonuclear diatomic molecules produce different rotational spectra,

even though the atoms are known to have identical chemical properties. This leads to the conclusion that the atoms must be

(a) isotopes, i.e. with the same atomic number.

(b) isobars, i.e. with the same atomic weight.

(c) isotones, i.e. with the same neutron number.

(d) isomers, i.e. with the same atomic number and weight.

23.

In the generalized operational amplifier circuit shown on the right, the op. amp. has a very high input impedance ( M ) and an open gain of 1000 for the frequency range under consideration. Assuming that the op. amp. draws negligible current, the voltage ratio is approximately

(a) (b) (c) (d)

24.

In the transistor circuit shown on the right, assume that the voltage drop between the base and the emitter is 0.5 V. What will be the ratio of the resistances , in order to make this circuit function as a source of constant current, ?

(a) (b) (c) (d)

k k

k

V

LOAD

9

25. For the circuit depicted on the right, the input

voltage is a simple sinusoid as shown below, where the time period is much smaller compared to the time constant of this circuit.

The voltage across is best represented by

(a)

(b)

(c)

(d)

DO NOT WRITE IN THIS SPACE

10

GS-2016-X (Physics)

Section B To be attempted only by candidates for Integrated Ph.D. programme.

(Candidates for Ph.D. programme will get NO credit for attempting this section.)

26. The integral

evaluates to

(a) zero (b) (c) (d)

27. A uniform solid wheel of mass and radius is halted

at a step of height h as shown in the figure. The minimum force , applied horizontally at the centre of the wheel, necessary to raise the wheel over this step is

(a) (b)

(c) (d)

28. In a futuristic scenario, two spaceships, A and B, are running a race, where

they start from the same point (marked START) but fly in opposite directions at constant speeds close to the speed of light. An observer fixed at the starting point observes that they both cross the points marked END, which are equidistant from the starting point, at the same time. Afterwards this observer receives messages from both spaceships.

Which of the following could be true?

(a) Both A and B agree that A won the race.

(b) A and B both claim to have won the race.

(c) Both A and B agree that they crossed the end point simultaneously.

(d) A thinks B won the race while B thinks A won the race.

11

29. The equation of state for a gas is given by

where and represent pressure, volume, temperature, number of atoms and the Boltzmann constant, respectively, while and are constants specific to the gas. If the critical point C corresponds to a point of inflexion of the - curve, then the critical volume and critical pressure for this gas are given by

(a) (b)

(c) (d)

30.

The lattice constant of a material is of the order of a , and its bond energies are of the order of an eV. The bulk modulus of such a material, in Pascals, is of the order of

(a) (b) (c) (d)

31.

Two harmonic oscillators A and B are in excited eigenstates with the same excitation energy E, as measured from their respective ground state energies. The natural frequency of A is twice that of B.

If the wavefunction of B is as sketched in the above picture, which of the following would best represent the wavefunction of A ?

(a)

(b)

(c)

(d)

12

32. Three positively charged particles lie on a straight line at positions and as indicated in the figure below. Their charges are and cmrespectively.

If the charges at and are fixed and the charge at is movable, the system will be in equilibrium when

(a) (b) (c) (d)

33. Consider the following system. Two circularloops of wire are placed horizontally, having acommon axis passing vertically through thecentre of each coil (see figure). The lower loophas radius and carries a current as shown inthe figure. The upper loop has a radius ( )and is at a distance ( ) above it.

If the lower loop is held fixed and the upper loop moves upwards with a uniform velocity

, then the induced e.m.f. and the direction of the induced current in this loop will be

(a) ; anti-clockwise (b) ; clockwise

(c) ; anti-clockwise (d) ; clockwise

34. A student in the laboratory is provided with a bunch of standard resistors aswell as the following instruments

Voltmeter accurate to 0.1 V Ammeter accurate to 0.01 A Constant current source (ideal)

Thermometer accurate to 0.1 C Stop watch accurate to 0.05 s Constant voltage source (ideal)

Using this equipment (and nothing else), the student is expected to measure the resistance of one of the given resistors. The least accurate result would be obtained by

(a) measuring the Joule heating.

(b) passing a constant current and measuring the voltage across it.

(c) measuring the current on application of a constant voltage across it.

(d) the Wheatstone bridge method.

13

35. In a digital circuit for three input signals (A, B and C) the final output (Y)should be such that for inputs

A B C 0 0 0 0 0 1 0 1 0

the output (Y) should be low and for all other cases it should be high.

Which of the following digital circuits will give such output?

(a) (b)

(c) (d)

PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER The following questions (36 – 40) must be answered by integers of 3 digits each. You may round off, e.g. as and e.g. as

and so on. Answer these questions on the OMR by filling in bubbles as you did for your reference code. Use only values of constants given in the table ‘USEFUL

CONSTANTS’.

Note that if the answer is, e.g. 25, you must fill in 025 and if it is, e.g. 5, you must fill in 005. If it is 0, you must fill in 000. If the zeros are not filled in (where required), the answer will be not be credited.

There are NO NEGATIVE MARKS for these questions.

36. On a planet having the same mass and diameter as the Earth, it is observed that objects become weightless at the equator. Find the time period of rotation of this planet in minutes (as defined on the Earth). Answer = 085

14

37. The function satisfies the differential equation

with the initial condition . What will be the value of ?

38. Two containers are maintained at the same temperature and are filledwith ideal gases whose molecules have mass and respectively. Themean speed of molecules of the second gas is times the r.m.s. speed ofthe molecules of the first gas. Find the ratio of , to the nearestinteger.

39. A particle is confined in a one-dimensional boxof unit length, i.e. , i.e. in a potential

The energy eigenvalues of this particle are denoted , , , , ...

In a particular experiment, the wavefunction of this particle, at , is given by

If, simultaneously, i.e. at , a measurement of the energy of the particle is made, find , where is the probability that the measurement will yield the energy .

40. Consider a sawtooth waveform which rises linearly from Volt to Volt in ns and then decays linearly to V over a period of ns. Find the r.m.s.voltage in units of milliVolt?


Answer = 081

Answer = 118

Answer = 000

Answer = 577

15

GS-2016-X (Physics)

Section C To be attempted only by candidates for Ph.D. programme.

(Candidates for Integrated Ph.D. programme will get NO credit for attempting this section.)

41. The value of the integral

where is the circle of centre z = 0 and radius =1

(a) (b) (c) (d)

42. In a Rutherford scattering experiment, the number of particles scattered ina direction , i.e. , as a function of the scattering angle (in thelaboratory frame) varies as

(a) (b)

(c) (d)

43. In a simple stellar model, the density of a spherical star of mass variesaccording to the distance from the centre according to

where is the radius of the star. The gravitational potential energy of this star (in terms of Newton’s constant ) will be

(a) (b)

(c) (d)

44. A particle moving in one dimension is confined inside a rigid box locatedbetween and . If the particle is in its ground state

the quantum mechanical probability of its having a momentum is given by

(a) (b)

(c) (d)

16

45. Consider two spin-1/2 identical particles A and B, separated by a distance ,interacting through a potential

where is a positive constant and the spins are in terms of the Pauli spin matrices. The expectation values of this potential in the spin-singlet and triplet states are

(a) (b)

(c) (d)

46. Two semi-infinite slabs A and B of dielectric constant and meet in aplane interface, as shown in the figure below.

If the electric field in slab A makes an angle with the normal to the boundary and the electric field in slab B makes an angle with the same normal (see figure), then

(a) (b)

(c) (d)

47. The Weizsäcker semi-empirical mass formula for an odd nucleus with protons and nucleons may be written as

where the are constants independent of . For a given , if is the number of protons of the most stable isobar, the total energy released when an unstable nuclide undergoes a single decay to ( ) is

(a) (b) (c) (d)

A B

17

48. In the experiment shown in figure (i) below, the emitted electrons from thecathode (C) are made to pass through the mercury vapor filled in the tube byaccelerating them using a grid (G) at potential V, positive w.r.t. the cathode.The electrons are collected by the anode (A).

(i)

Voltage

(ii)

The variation of electron current (I) as a function of V is given in figure (ii). The shape of this curve must be interpreted as due to

(a) ionization of mercury atoms.

(b) an emission line from mercury atoms.

(c) attachment of electrons to mercury atoms.

(d) resonant backscattering of electrons to cathode from grid.

49. The dispersion relation for electrons in the conduction band of a -typesemiconductor has the form where and are constants. Itwas observed that the cyclotron resonance frequency of such electrons is

rad s-1, when placed in a magnetic field W m-2. It followsthat the constant must be about

(a) (b) (c) (d)

50. Consider the hyperon decay followed by . If theisospin component, baryon number and strangeness quantum numbers aredenoted by , and respectively, then which of the following statements iscompletely correct?

(a) In (1) is not conserved, is conserved, is not conserved;In (2) is conserved, is conserved, is conserved.

(b) In (1) is conserved, is not conserved, is not conserved; In (2) is conserved, is conserved, is conserved.

(c) In (1) is not conserved, is conserved, is not conserved; In (2) is not conserved, is conserved, is conserved.

(d) In (1) is not conserved, is conserved, is conserved; In (2) is conserved, is conserved, is conserved.

18

PLEASE READ CAREFULLY BEFORE PROCEEDING FURTHER The answer to the following questions (51 – 55) must be answered by integers of 3 digits each. You may round off, e.g. as and e.g.

as and so on. Answer these questions on the OMR by filling in bubbles as you did for your reference code. Use only values of constants given in the table ‘USEFUL

CONSTANTS’.

Note that if the answer is, e.g. 25, you must fill in 025 and if it is, e.g. 5, you must fill in 005. If it is 0, you must fill in 000. If the zeros are not filled in (where required), the answer will be not be credited.

There are NO NEGATIVE MARKS for these questions.

51. Given the infinite series

find the value of for .

52. A quantum mechanical plane rotator consists of two rigidly connectedparticles of mass m and connected by a massless rod of length d is rotating inthe x-y plane about their centre of mass. Suppose that the initial state of therotor is given by

where is the angle between one mass and the x axis, while is a normalization constant. Find the expectation value of in this state, in units of .

53. A continuous monochromatic ( ) laser beam is chopped into0 pulses using some sort of shutter. Find the resultant linewidth ofthe beam in units of .

54. N distinguishable particles are distributed among three states havingenergies and respectively. If the total equilibrium energy ofthe system is , find the number of particles.

Answer = 343

Answer = 004

Answer = 012

Answer = 325

19

55. The circuit shown below contains an unknown device X.

The current-voltage characteristics of the device X were determined and are shown in the plot given below.

Determine the current I (in mA) flowing through the device X.


2 V

100

100

Answer = 012

20


fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES


Head office fiziks, H.No. 40 D, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498

Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16

TIFR GS-2017 (Physics)

Section A

Q1. Denote the commutator of two matrices A and B by ,A B AB BA and the

anticommutator by ,A B AB BA .

If , 0A B , we can write ABC

(a) ,B A C (b) ,B A C (c) ,A C B (d) ,B A C

Q2. Consider the waveform x t shown in the diagram below.

The Fourier series for x t which gives closest approximation to this waveform is

(a) 2 1 4 1 3cos cos cos ...

2 3

t t tx t

T T T

(b) 2 2 1 4 1 6cos cos cos ...

2 3

t t tx t

T T T

(c) 2 1 4 1 3sin sin sin ...

2 3

t t tx t

T T T

(d) 2 1 2 1 3sin sin sin ...

2 3

t t tx t

T T T

x

2T T

1

0

1

T 2T 3Tt





Q3. A solid tetrahedron (solid with four plane sides) has the following projections (drawn to

scale) when seen from three different sides:

When viewed from thee front, its projection will be

(a) (b)

(c) (d)

Q4. A small elastic ball of mass m is placed at the apex of a 045 inclined plane as shown in

the figure below.

The ball is allowed to slip without friction down the plane (along the dotted line), hit the

ground (as shown) and bounce along it. If the height of the inclined plane is h and the

coefficient of restitution between the ball and the ground is 0.5 , then the distance AB , as

marked on the figure, will be

(a) 1 2 h (b) 3 2h (c) 3h (d) 2h

Left Top Right

A B





Q5. Two masses 3m and 2m are suspended vertically by identical massless

springs, each of stiffness constant k . The mass 2m is suspended from the

mass 3m and the mass 3m is suspended from a rigid support, as shown in

the figure. If only vertical motion is permitted, the frequencies of small

oscillations of this system are

(a) 3

,2

k k

m m (b) ,

2 3

k k

m m

(c) ,6

k k

m m (d)

2 3,

3 2

k k

m m

Q6. Two long hollow conducting cylinders, each of height h , are placed

concentrically on the ground, as shown in the figure (top view). The

outer cylinder is grounded, while the inner cylinder is insulated. A

positive charge (the black dot in the figure) is placed between the

cylinders at a height 2

h from the ground. Which of the following figures gives the most

accurate representation (top view) of the lines of force?

(a) (b)

(c) (d)

k

k

3m

2m

0V

0V

0V

0V 0V





Q7. A common model for the distribution of charge in a hydrogen atom has a point-like

proton of charge 0q at the centre and an electron with a static charge density

distribution

2

03

r

aq

r ea

where a is a constant. The electric field E

at r a due to this system of charges will be

(a) 02 2

0

5ˆ

4

qr

e a

(b) 0

20

5ˆ

4

qr

e a

(c) 02 2

0

3ˆ

4

qr

e a (d) 0

2 20

5ˆ

4

qr

e a

Q8. A rectangular metallic loop with sides 1L and 2L is placed in

the vertical plane, making an angle with respect to the x -

axis, as shown in the figure, and a spatially uniform magnetic

field ˆB By

is applied. The loop is free to rotate about the z

axis (shown in the figure with a double line). The magnetic

field changes with time at a constant rate

dB

dt

If the resistance of the loop is R , the torque required to prevent the loop from rotating

will be

(a) 2

1 2 ˆsin2

L LB z

R (b)

2

1 2 ˆsin 22

L LB z

R

(c) 2

1 2 ˆsin cosL L

B zR

(d) 2

1 2 ˆsinL L

B zR





Q9. Consider the 1 D asymmetric double well potential V x as sketched below.

The probability distribution p x of a particle in the ground state of this potential is best

represented by

(a) (b)

(c) (d)

Q10. The normalized wave function of a particle can be written as

0

1

7

n

nn

x N x

where n x are the normalized energy eigenfunctions of a given Hamiltonian. The

value of N is

(a) 6 2 7

7

(b)

1

7 (c)

3

7 (d)

6

7

V x

x

p x

x

p x

x

p x

x

p x

x





Q11. Consider a system of non-interacting particles with integer angular momentum J at a

temperature T . This system is placed in a magnetic field B in the z direction. The

energy of a state with zJ mh is

m BE m B

with 0B . The fractional magnetization of the particles as a function of B

B

B

k T

can be

represented as

(a) (b)

(c) (d)

Q12. The separation between neighbouring absorption lines in a pure rotational spectrum of the

hydrogen bromide HBr molecule is 2.23meV . If this molecule is considered as a rigid

rotor and the atomic mass number of Br is 80 , the corresponding absorption line

separation in deuterium bromide DBr molecule, in units of meV ,would be

(a) 2.234 (b) 1.115 (c) 4.461 (d) 1.128

0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.80.0

/B kT

m/J

0.60.70.80.91.0

0.5 1.0 1.5 2.00.0 /B kT

m/J

0.50.40.3

0.8

0.6

0.4

0.2

0.00.5 1.0 1.5 2.00.0

/B kT

m/J

0.70.60.50.40.3

0.5 1.0 1.5 2.00.0 /B kT

m/J

0.80.91.0





Q13. Consider a 2 D square lattice. The ratio of the kinetic energy of a free electron at a

corner of the first Brillouin zone cE to that of an electron at the midpoint of a side

face-of the same zone mE is c

m

E

E

(a) 2 (b) 2 (c) 1 (d) 1

2

Q14. A current source produces a square wave I t of 1.0V peak-to-peak voltage and is used

to drive the RC circuit shown below.

Which of the following represents the correct voltage across the capacitor C ?

(a) (b)

(c) (d)

Q15. The output Y of the following circuit will be

(a) C (b) B (c) A (d) A B C

R V t

C

I t

A A B B C C

Y





Q16. A space telescope in orbit around the Earth discovers a new planet, which is observed to

move around the Sun by an angle of 4.72 milliradians in a year. Assuming a circular

orbit, estimate the distance, in A.U., of the planet front the Sun.

Q17. The matrix

100 2 0

0

0 100 2

x

x x

x

where 0x , is known to have two equal eigenvalues. Find the value of .x

Q18. A beam of plane microwaves of wavelength 12cm strikes the surface of a dielectric at

045 . If the refractive index of the dielectric is 4

3, what will be the wavelength, in units

of mm , of the microwaves inside the dielectric?

Q19. A system of particles occupying single-particle levels and obeying Maxwell Boltzmann

statistics is in thermal equilibrium with a heat reservoir at temperature T . If the

population distribution in the non-degenerate energy levels is as shown in the table below,

what would be the temperature of the system in degree Kelvin?

Energy eV Population %

30.30 3.16

21.60 8.69

13.01 23.54

4.31 64.61

Q20. A thermally isolated container stores 2N gas at 027.24 C at one atmospheric pressure.

Suddenly the pressure of the gas is increased to two atmospheric pressures. Assuming

2N to behave as an ideal gas, estimate the change in temperature of the gas, in Celsius

degrees ( 0C ).

Q21. A signal is to be sent from a coaxial cable with impedance 40 into a second coaxial

cable with impedance 60 . We can prevent reflection at the joint between the cables, by





adding an impedance in parallel to the second cable. What should be the value, in units of

Ohms , of that impedance?

Q22. An AC voltage source has an internal resistance of 50 and is specified to deliver an

rms voltage of 50V to a matched load. If you connect this AC source to a cathode- ray

oscilloscope with 1M input setting, what will be the peak-to-peak voltage you observe?

Q23. The energy of an electron in the ground state of the He atom is 79eV . Considering the

Bohr model of the atom, what would be 10 times the first ionization potential for a He

ion, in units of eV ?

Q24. A quantum mechanical system consists of a one-dimensional infinite box, as indicated in

the figures below.

3 (three) identical non-interacting spin-1/ 2 particles; are first placed in the box, and the

ground state energy of the system is found to be 0 18E eV . If 7 (seven) such identical

particles are placed in the box, what will be the ground state energy, in units of eV ?

Q25. Cosmic ray muons, which decay spontaneously with proper lifetime 2.2 s , are produced

in the atmosphere, at a height of 5km above sea level. These move straight downwards at

98% of the speed of light.

Find the percent ratio 100 A

B

N

N

of the number of muons measured at the top of two

mountains A and B , which are at heights 4,848m and 2,682m respectively above

mean sea level.

SECTION B

Q26. A unitary matrix U is expanded in terms of a Hermitian matrix H , such that

0E





2

i H

U e

if we know that

1 30

2 20 1 0

3 10

2 2

H

then U must be

(a)

30

2 20 0

30

2 2

i i

i

i i

(b)

1 3

2 21 1

2 2

3 1

2 2

i

i

i

(c)

1 0 3

0 2 0

3 0 1

(d)

32 1

21 2 0

30 2

2

i

i

i

Q27. A liquid is flowing through a capillary tube of inner radius r under the influence of an

external pressure P . The uncertainties in the measurements of P and r are found to be

2% and 1% , respective1y. The uncertainty in the flow of liquid per second is

(a) 3.61% (b) 2.23% (c) 2.83% (d) 4.47%

Q28. A uniform solid sphere 1S of radius r and mass m is rolling without slipping on top of

another sphere 2S of radius R , as shown in the figure. Initially

2S

R

N1Sr

P





1S was at rest directly on top of 2S , and then it started rolling down under the influence

of gravity. The point of contact P subtends an instantaneous angle from the topmost

point N of the lower sphere at the centre of the lower sphere. At what minimum value of

will the spheres lose contact?

(a) 1 5cos

13 (b) 1 5

cos12

(c) 1 2cos

3 (d) 1 12

cos13

Q29. An electromagnetic wave in free space is described by

0

1ˆ, , , cos 3 2

2E x y z t z E kx ky t

The Poynting vector associated with this wave is along the direction

(a) ˆ ˆ3x y (b) ˆ ˆ3x y (c) ˆ ˆ3x y (d) ˆ ˆ3x y

Q30. Electrons in a given system of hydrogen atoms are described by the wave function

/3100 311, , 0.8 0.6 ir e

where the n m denote normalized energy eigenstates. If ˆ ˆ ˆ, ,x y zL L L are the components

of the orbital angular momentum operator, the expectation value of 2ˆxL in this system is

(a) 21.5 (b) 20.18 (c) 20.36 (d) Zero

Q31. In two dimensions, two metals A and B , ha the number density of free electrons in the

ratio

: 1: 2A Bn n

The ratio of their Fermi energies is

(a) 2 : 3 (b) 1:8 (c) 1: 2 (d) 1: 4

Q32. One mole of monoatomic ideal gas is initially at pressure 0P

and volume 0V .

The gas then undergoes a three-stage cycle consisting of- the

following processes 0

1

2P

0P

P

0T

0T

V02V0V

0

1

2T





(i) An isothermal expansion till it reaches volume 02V , and heat Q flows into the gas

(ii) An isobaric compression back to the original volume 0V

(iii) An isochoric increase in pressure till the original pressure 0P is regained.

The efficiency of this cycle can be expressed as

(a) 0

0

4 2

4 3

Q RT

Q RT

(b) 0

0

4 2

4 3

Q RT

Q RT

(c) 0

0

4 2

4

Q RT

Q RT

(d) 0

0

4 2

4

Q RT

Q RT

Q33. A deuteron of mass M and binding energy B is struck by a gamma ray photon of energy

E and is observed to disintegrate into a neutron and a proton. If 2B Mc , the minimum

value of E must be

(a) 2

22

2

BB

Mc (b)

2

2

13

2

BB

Mc

(c) 2

2

BB

Mc (d)

2

2

12

2

BB

Mc

Q34. Light passes through a narrow slit and gives the Fraunhofer diffraction pattern shown in

the adjacent figure.

Which of the following could be the shape of the slit?

(a) (b)





(c) (d)

Q35. For exact calculation and minimum complexity, two four-digit binary numbers can be

added with

(a) 3 full adders and 1 half-adder (b) 2 full adders and 2 half-adders

(c) 1 full adder and 3 half-adders (d) 4 full adders

Q36. Which digital logic gate is mimicked by the following silicon diode and silicon transistor

circuit?

Q37. Consider the following situations.

A

B2R

3R

1CR2CR

1R

outV

5V5V

A

d q

d

B

d q





In situation A , two semi-infinite earthed conducting planes meet at right-angles. A point

charge q , is placed at a distance d from each plane, as shown in the figure A . The

magnitude of the force exerted on the charge q is denoted AF .

In situation B , the same charge q is kept at the same distance d from an infinite earthed

conducting plane, as shown in the figure B . The magnitude of the force exerted on the

charge q is denoted BF . Find the numerical ratio A

B

F

F

Q38. Two identical bosons may occupy any of two energy levels 0, , where 0 . The

lowest energy state is doubly-degenerate and the excited state is non-degenerate.

Assume that the two-particle system is in thermal equilibrium at a temperature T .

Calculate the average energy E . What will be the leading term of

expB

E

k T

at low temperature?

Q39. Evaluate the expression

1 2 3 2 1

21 2 3... 1 0

0 0 0 0 0 0

!n nx x x x xA

n n nn dx dx dx dx dx dx

Q40. In outer space, where the effects of gravity can be neglected, a drop of

liquid assumes a spherical shape. However, when disturbed it

undergoes shape oscillations (see figure). The frequency v of

oscillation of a drop depends on its equilibrium radius, its density and

the surface tension.





What would be the numerical value of the ratio 2

Hg

H O

v

vof the frequencies of oscillation

between a drop of mercury Hg and a drop of water 2H O of the same equilibrium

radius?

You may use the following data:

Liquid Density in gm 3cm Surface tension in 1Nm

Water 1.0 0.073

Mercury 13.6 0.487

Q41. The value of the integral

40 4

dx

x

is

(a) 8

(b)

4

(c)

2

(d)

Q42. The Lagrangian of a system described by a single generalised coordinate q is

21sin

2L q q

Its Hamiltonian is

(a) zero (b) 21sin

2q p q

(c) 2sinq q (d) not defined

Q43. A one-dimensional quantum harmonic oscillator of natural frequency is in thermal

equilibrium with a heat bath at temperature T . The mean value E of the energy of the

oscillator can be written as

(a) coth2 2 Bk T

(b) csch

2 2 Bk T





(c) sech2 2 Bk T

(d) tanh

2 2 Bk T

Q44. Consider a spherical shell with radius R such that the potential on

the surface of the shell in spherical coordinates is given by,

20, , cosV r R V

where the angle is shown in the figure. There are no charges

except for those on the shell. The potential outside the shell at the

point P a distance 2R away from its center C (see figure) is

(a) 20 1 2cos8

VV (b) 20 1 cos

4

VV

(c) 20 1 cos8

VV (d) 30 2cos cos

2

VV

Q45. A quantum mechanical system which has stationary states 1 , 2 and 3 , corresponding

to energy levels 0 ,1eV eV and 2eV respectively, is perturbed by a potential of the form

1 3 3 1V

where, in ,0 1eV .

The new ground state, correct to order , is approximately.

(a) 1 1 32 2

(b) 1 2 32

(c) 1 32

(d) 1 3

2

Q46. Hydrogen atoms in the atmosphere of a star are in thermal equilibrium, with an average

kinetic energy of 1eV . The ratio of the number of hydrogen atoms in the 2nd excited

state 3n to the number in the ground state 1n is

(a) 65.62 10 (b) 113.16 10

(c) 83.16 10 (d) 81.33 10

C

R

P





Q47. A photomultiplier tube is used to detect identical light pulses each of which consists of a

fixed number of photons The photoelectric efficiency is 10% , i.e. a photon has 10%

probability of causing the emission of a detectable photo-electron.

The photomultiplier gain is 610 .

The typical output current, as a function of time, is shown by the figure below for a few

pulses, where maxI is 80 A . It follows that the number of photons in each pulse is

(a) 65 10 (b) 50 (c) 800 (d) 5

Q48. A subatomic particle and its excited state * have rest masses 23.1 /GeV c and

23.7 /GeV c respectively. A table of its assigned quantum numbers is given below.

Angular Momentum Parity C-Parity Isospin Electric charge

1J 1P 1C 0I 0Q

If 0* is an excited state of 0 with a mass of about 21.3 /GeV c , which of the following

reactions is possible when the above quantum numbers are conserved ?

(a) * (b) * 0 0

(c) * (d) * 0*

Q49. The cosmic microwave background radiation in the Universe has a blackbody

distribution corresponding to a temperature 2.735 K . In a certain cosmological model, it

was assumed that the universe consists purely of radiation and is undergoing adiabatic

expansion. In this model it was predicted that the volume of the Universe will be tripled

in the next 1010 yrs. The corresponding black body radiation temperature would be

(a) 0.9116 K (b) 2.078 K (c) 1.526 K (d) 1.896 K

Q50. The following circuit is fed with an input sine wave of frequency50 Hz .

20 ns

tmaxI

I t

10 pF

10k

outV

10k

inV





Which of the following graphs (solid line is input and dashed line is output) best

represents the correct situation?

(a) (b)

(c) (d)

Q51. Electrons in a metal are scattered by both impurities and phonons. The impurity

scattering time is 128 10 s and the phonon scattering time is 122 10 s . Taking the

density of electrons to be 14 33 10 m , find the conductivity of the metal in units of

1 1AV m . [Assume that the effective mass of the electrons is the same as that of a free

electron.]

Q52. A particle of mass m , confined to one dimension x , is in the ground state of a harmonic

oscillator potential with a normalized wave function





2

1

4

0

2 axax e

where 2

ma

. Find the expectation value of 8x in terms of the parameter a

Q53. Write down x t , where x t is the solution of the following differential equation

2 1 1d d

xdt dt

,

with the boundary conditions

0

0

10,

2tt

dxx t

dt

Q54. Assume that the crystal structure of metallic copper Cu results in a density of

atoms 28 38.46 10cu m . Each Cu atom in the crystal donates one electron to the

conduction band, which leads, for the 3 D Fermi gas, to a density of states

3

1* 22

2 2

1 2

2

mg

where *m is the effective mass of the conduction electrons. In the low temperature limit

(i.e. 0T K ), find the Fermi energy FE , in units of eV . You may assume *m to be

equal to the free electron mass em .

Q55. In a theoretical model of the nucleus, the binding energy per nucleon was predicted as

shown in the figure below

00 50 100 150 200 250

1

2

3

4

5

6

7

8

9

Bin

ding

ene

rgy

per

nucl

eon

(BE/A

) (M

eV)

Mass Number (A)





If a nucleus of mass number 240A undergoes a symmetric fission to two daughter

nuclei each of mass number 120A , write down the amount of energy released in this

process, in units of MeV , using this theoretical model.

fiziks Institute for NET/JRF, GATE, IIT‐JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics

H.No. 40‐D, Ground Floor, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐110016

Phone: 011‐26865455/+91‐9871145498

Website: www.physicsbyfiziks.com | Email: [email protected]

1


GS-2018-X (PHYSICS)

SECTION A

Q1.

Refer to the figure above. If the z-axis points out of the plane of the paper towards you,

the triangle marked ‘A’ can be transformed (and suitably re-positioned) to the triangle

marked ‘B’ by

(a) Rotation about z - direction by 2, then reflection in the yz -plane

(b) Reflection in the xz - plane, then rotation by 2 about z-direction

(c) Reflection in the yz - plane, then rotation by 2 about z-direction

(d) Rotation about x - direction by 2, then rotation by 2 in the yz -plane

Q2. If a 2 2 matrix M is given by

1 1 2

1 2 0

iM

i

Then det exp M

(a) 2e e (b) e (c) 2 sin 2i (d) exp 2 2

B

z x

y

A

1

23



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2

Q3. A small block B of mass m is placed on an

inclined plane ,P which makes an angle with a

horizontal cart ,C on which P is rigidly fixed (see

figure). The coefficient of friction between the

block B and the plane C is .

When the cart stays stationary the block slides down. If the cart C is moving in the

horizontal direction with acceleration ,a the minimum value of a for which the block

remain static on the place is:

(a) tan

tan 1g

(b) sin cosg

(c) 1 tan

tang

(d) cos sing

Q4. A particle of mass m moving in one-dimension x is subjected to the Lagrangian

21

2L m x x

Where is a real constant. It is starts at the origin at 0,t its motion corresponds to the

equation ( a is a constant)

(a) expx a t (b) sinx a t

(c) 1 expx a t (d) sinhx a t

P

B

Ca



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3

Q5. The sketch below shows a radio antenna located at the edge of a calm lake, which has a

receiver R at the centre of the dish at a height 2a above the ground. This is picking up

a signal from a distant radio-emitting star which is just rising above the horizon.

However, the receiver also picks up a reflected signal from the surface of the lake, which,

at the relevant radio-wavelength, may be taken to be a plane.

If the instantaneous angle of the star above the horizon is denoted , the receiver R will

detect the first interference maximum when

(a) arcsin1 3

2a

(b) arcsin1 3

a

(c) arcsin2a

(d) arcsin

a

Q6. A particle is confined inside a one-dimensional box of length and left undisturbed for a

long time. In the most general case, its wave-function MUST be

(a) The ground state of energy.

(b) Periodic, where equals an integer number of periods.

(c) Any one of the energy eigenfunctions.

(d) A linear superposition of the energy eigenfunctions.

Q7. A classical ideal gas of atoms with masses m is confined in a three-dimensional potential

2 2 2, ,2

V x y z x y z

at a temperature .T If Bk is the Boltzmann constant, the root mean square (r.m.s.)

distance of the atoms from the origin is

(a) 1 2

3 Bk T

(b) 1 2

2

3Bk T

(c) 1 2

3

2Bk T

(d) 1 2

Bk T

a

R

2

a



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4

Q8. The characteristic impedance of a co-axial cable is independent of the

(a) Dielectric medium between the core and the outer mesh

(b) Outer diameter

(c) Length of the cable

(d) Core diameter

Q9. The figure below shows an unknown circuit, with an input and output voltage signal.

From the form of the input and output signals, one can infer that the circuit is likely to be

(a) (b)

(c) (d)

Q10. In Boolean terms, A B A C is equal to

(a) ABC (b) A BC

(c) A B C (d) A B C A B

Q11. Consider the two equations

2 2

13 2

x y 3 1x y

How many simultaneous real solutions does this pair of equations have?

UNKNOWN

CIRCUIT

INPUT OUTPUT



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5

Q12. A heavy steel ball is struck by a small steel pellet moving horizontally with velocity

120 .ms If the pellet bounces off the steel ball with no slippage, and then rises vertically

to a height 10m above the point of contact, then what is 100 times the elastic coefficient

of restitution e i.e. 100 ?e

Q13. A particle is in the ground state of a cubical box of side . Suddenly one side of the box

changes from to 4 . If p is the probability of finding the particle in the ground state of

the new box, what is 1000 ?p

Q14. The wave-function of a particle in a one-dimensional harmonic oscillator potential is

given by

1 4 2

2 2

1 21 exp

2

x x

Where 100 .m Find the expectation value of the position x of this particle, in .m

Q15. Consider a dipole antenna with length , charge q and frequency . The power emitted

by the antenna at a large distance r is .P Now suppose the length is increased to 2 ,

the charge is increased to 3q and the frequency is increased to 5 . By what factor is

the radiated power increased?

Q16. Calculate the self-energy, in Joules, of a spherical conductor of radius 8.5 ,cm which

carries a charge 100 .C

Q17. A heat engine is operated between two bodies that are kept at constant pressure. The

constant-pressure heat capacity pc of the reservoirs is independent of temperature.

Initially the reservoirs are at temperatures 300K and 402 .K If, after some time, they

come to a common final temperature ,fT the process remaining adiabatic, what is the

value of fT (in Kelvin)?



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6

Q18. N Particles are distributed among three energy levels having energies: 0, Bk T and

2 Bk T respectively. If the total equilibrium energy of the system is approximately

42.5 Bk T then find the value of N (to the closest integer).

Q19. A realistic voltmeter can be modeled as an ideal voltmeter with an input resistor in

parallel a shown below:

Such a realistic voltmeter, with input resistance 1 ,k gives a reading of 100mV when

connected to a voltage source with source resistance 50 . .What will a similar voltmeter,

with input resistance 1 ,M read in ,mV when connected to the same voltage source?

Q20. An electron enters a linear accelerator with a speed 110 .v ms A vertical section of the

accelerator tube is shown in the figure, where the lengths of the successive sections are

designed such that the electron takes the same time 20ms to traverse each section.

If the momentum of the electron increases by 2% every time it crosses the narrow gap

between two sections, what is the length (in km) of the collider which will be required to

accelerate it to 1100 ?kms



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7

SECTION B

Q21. If y x satisfies the differential equation 4 4 0y y y with boundary conditions

0 1y and 0 0,y then 1

2y

(a) 2

e (b)

1 1

2e

e

(c) 1

e (d)

2

e

Q22. Given the following xy data

Which of the following would be the best curve, with constant positive parameters a and

,b to fit this data?

(a) y ax b (b) expy a bx (c) 10logy a bx (d) expy a bx

Q23. The Hamiltonian of a dynamical system is equal to its total energy, provided that its

Lagrangian

(a) Does not contain velocity-dependent terms.

(b) Has no explicit time dependence.

(c) Is separable in generalized coordinates and velocities.

(d) Does not have terms which explicitly depend on the coordinates.

Q24. A particle in a one-dimensional harmonic oscillator potential is described by a wave-

function , .x t If the wave function changes to ,x t then the expectation value of

kinetic energy T and the potential energy V will change, respectively, to

(a) 2 2and T V (b) 2 2and T V

(c) 2 2and T V (d) 2 2and T V

1.0 2.0 3.0 4.0 5.0

0.002 0.601 0.948 1.21 1.42

x

y



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Q25. An electron is in the 2s level of the hydrogen atom, with the radial wave-function

3 2

0 00

12 exp .

22 2

r rr

a aa

The probability P r of finding this electron between distances r to r dr from the

centre is best represented by the sketch

(a) (b)

(c) (d)

Q26. An atom of atomic number Z can be modeled as a point positive charge surrounded by a

rigid uniformly negatively charged solid sphere of radius .R The electric Polaris ability

a of this system is defined as

Ep

E

Where pp is the dipole moment induced on application of electric field E which is small

compared to the binding electric field inside the atom. It follows that a

(a) 03

4

R

(b) 0

3

8

R

(c) 3

04 R (d) 308 R

P r

r

P r

r

P r

r

P r

r



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9

Q27. A many-body system undergoes a phase transition between two phases A and B at a

temperature cT The temperature-dependent specific heat at constant volume VC of the

two phases are given by 3AVC aT bT and 3.B

VC cT Assuming negligible volume

change of the system, and no latent heat generated in the phase transition, cT is

(a) 4b

c a (b)

3b

c a (c)

2b

c (d)

b

c a

Q28. Consider the following circuits C-1 and C-2.

You can apply the golden rules of an ideal op-amp to

(a) Both C-1 and C-2 (b) Neither C-1 nor C

(c) Only C-1 (d) Only C-2

Q29. The current i flowing through the following circuit is

(a) 1.0A (b) 0.75A (c) 0.6A (d) 0.5A

9

9

9

9

43

3

33

3

3

6

3V

i

C-1

C-2



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10

Q30. Two students A and B try to measure the time period T of a pendulum using the same

stopwatch, but following two different methods. Student A measures the time taken for

one oscillation, repeats this process 1N times and computes the average. On the

over hand, Student B just once measures the time taken for N oscillations and divides

that number by .N

The error in the measurement of the stopwatch is much smaller than the time period.

Assuming that accuracy of the stopwatch is sufficiently high which of the following

statements is true about the errors in T as measured by A and by ?B

(a) A and B will measure the time period with the same accuracy.

(b) It is not possible to determine if the measurement made by A or B has the larger

error.

(c) The measurement made by A has a smaller error than that made by .B

(d) The measurement made by A has a larger error than that made by .B

Q31. Evaluate the integral

2exp cos 2dx x x

Q32. From an observational post E on the Earth, two ballistic missiles, each of rest length

from nose-tip to tail-end, are observed to fly past each other, with the same uniform

relativistic speed 2,c in opposite directions, as shown below.

What is the time taken for the tail-end of one of the missiles to cross the tail-end of the

other missile, as measured from the post ?E

v

v



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11

Q33. A statistical system, kept at a temperature ,T has n discrete energy levels with equal

level-spacing , starting from energy 0. If, now, a single particle is placed in the system

what will be the mean energy of the system in the limit as ?n [The answer should

not be left as a summation]

Q34. Consider an infinite plane with a uniform positive charge density as shown below. A

negative point charge q with mass m is held at rest at a distance d from the sheet and

released. It will then undergo oscillatory motion. What is the time period of this

oscillation?

[You may assume that the point charge can move freely though the charged plane without

disturbing the charge density.].

Q35. Given a particle confined in a one-dimensional box between x a and ,x a a

student attempts to find the ground state by assuming a wave-function

3 22 2 for

0 for

A a x x a

x ax

The ground state energy mE is estimated by calculating the expectation value of energy

with this trial wave-function. If 0E is the true ground state energy, what is the ration

0 ?mE E

qd



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12

SECTION C

Q36. The Fourier series which reproduces, in the interval 0 1,x the function

n

f x x n

Where n is an integer, is

(a) 1 2cos 2 2cos 4 2cos 6x x x to

(b) 1 cos cos 2 cos3x x x to

(c) cos cos 2 cos3x x x to

(d) 1 1cos sin cos 2 sin 2 cos3 sin 3

2 3x x x x x x to

Q37. The value of the integral 2 2

1 cos x

x a

is.

(a) 1 2a (b) 1 2 a (c) exp a a (d) expa a

Q38. A dynamical system with one degree of freedom is described by canonical coordinate

, .p q The generator F of the canonical transformation , ,p q q p is

(a) F pq (b) F pq (c) F pq (d) F pq

Q39. The electrostatic charge density r corresponding to the potential

0

11 exp

4 2

q rr r

r

is

(a) 32 expq r q r (b) 3

exp4

q r q r

(c) 32 expq r q r (d) 3

exp2

aq r q ar



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13

Q40. The Hamiltonian of a particle of charge q and mass m in an electromagnetic field is

given by 21

, ,2

H p qA x t q x tm

.

Where , A

are the electromagnetic potentials? Clearly this Hamiltonian changes under

a gauge transformation.

x

t

A A

Where ,x t is a gauge function? Nevertheless the motion of the particle is not affected

because

(a) The action of the particle changes only by surface terms which do not vary.

(b) The Lorentz force is modified to balance the effect of the gauge transformation.

(c) The Lagrangian does not change under the gauge transformation.

(d) The motion of the particle is correctly described only in Lorenz gauge.

Q41. The electron of a free hydrogen atom is initially in a state with quantum numbers 3n

and 2. It then makes an electric dipole transition to a lower energy state. Which one

of the given states could it be in after the transition?

(a) 3, 0n (b) 2, 1n (c) 3, 1n (d) 2, 2n

Q42. Consider a monatomic solid lattice at a low temperature ,DT T where DT is the

characteristic Debye temperature of the solid D m BT k where m is the maximum

possible frequency of the lattice vibrations). The heat capacity of the solid is proportional

to

(a) DT T (b) DT T (c) 3

DT T (d) 2

DT T



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Q43. A signal 5sin 100inV t t is sent to both the circuits sketched below.

If the DC output voltage of the top circuit has a value 1dcV and the bottom circuit has a

value 2 ,dcV then which of the following statements about the relative value of 1dcV and

2dcV is correct?

(a) It will depend on the slew rate of the op-amp.

(b) 1 2dc dcV V

(c) 1 2dc dcV V

(d) 1 2dc dcV V

Q44. Consider the circuit shown on the right, which

involves an op-amp and two resistors, with an input

voltage marked INPUT.

Which of the following circuit components, when

connected across the input terminals, is most likely

to create a problem in the normal operation of the

circuit?

(a) A voltage source with very high Thevenin resistance.

(b) A voltage source with a very low Thevenin resistance.

(c) A current source with a very high Norton resistance.

(d) A current source with a very low Norton resistance.

inV t

1k10 F

1k1dcV~

inV t

1k

10 F

1k 1dcV~

INPUT



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Q45. Which one of the following circuits, constructed only with resistors and voltmeters, will

allow you to obtain the correct value of resistance sr using the voltmeter readings? Note

that the value of BR is known while 1 2 3 4, , ,r r r r and sr are all unknown.

[Assume that the voltmeters and resistors are ideal.]

(a) (b)

(c) (d)

Q46. A fourth rank Cartesian tensor ijktT satisfies the following identities

(i) ijk jikT T (ii) ijk ij kT T (iii) ijk k ijT T

Assuming a space of three dimensions (i.e. , , 1, 2, 3i j k , what is the number of

independent components of ?ijkT

Q47. If the velocity of the Earth in its orbit is ,v find ,E E where E is the translational

(non-relativistic) kinetic energy of the Earth and E is its relativistic correction to the

lowest order in .v c

BR 1r 2r

sr

3r 4r

BR 1r 2r

sr

3r 4r

BR 1r 2r

sr

3r 4r

BR 1r 2r

sr

3r 4r



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Q48. A plane electromagnetic wave, which has an electric field

ˆ ˆ, expz

E x t Pi Qj i tc

is passing through vacuum. Here ,P Q and are all constants, while c is the speed of

light in vacuo.

What is the average energy flux per unit time (in SI units) crossing a unit area placed

normal to the direction of propagation of this wave, in terms of the above constants?

Q49. The state of a spin -1particle is given by

1 2

1, 1 1, 0 exp 1, 1 exp3 33

i i

Where , sS M denote the spin eigenstates with eigenvalues 2 1h S S and shM

respectively. Find ,xS i.e. the expectation value of the x component of the spin.

Q50. A particle of mass m moves in a two-dimensional space ,x y under the influence of a

Hamiltonian.

2 2 2 2 21 15 5 6

2 4x yH p p m x y xym

Find the ground state energy of this particle in a quantum-mechanical treatment.


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Website: www.physicsbyfiziks.com | Email: [email protected] 1


GS 2019 ADMISSION TEST

Instructions for candidates for Integrated Ph.D. Programme in Physics

PLEASE READ THESE INSTRUCTIONS CAREFULLY BEFORE YOU ATTEMPT

THE QUESTIONS

This test consists of TWO sections.

SECTION A comprises 25 questions, numbered Q1 - Q25

These are questions on basic topics.

SECTION B comprises 15 questions, numbered Q1 - Q15

These may require somewhat more thought/knowledge.

ALL questions are Multiple-Choice Type. In each case, ONLY ONE option is correct.

Answer them by clicking the radio button next to the relevant option.

If your calculated answer does not match any of the given options exactly, you may mark the

closest one if it is reasonably close.

The grading scheme will be as follows:

Section A : +3 marks if correct; –1 mark if incorrect; 0 marks if not attempted

Section B : +5 marks if correct; 0 marks if incorrect or not attempted,

i.e. NO negative marks.




SECTION A

Q1. Which of the following operations will transform a tetrahedron ABCD with vertices as listed

below

x y z

A 0 0 0

B 1 0 0

C 0 1 0

D 0 0 2

Into a tetrahedron ABCD with vertices as listed below

x y z

A 0 0 0

B 0 1 0

C 0 0 1

D 2 0 0

Up to suitable translation?

(a) A reflection in the xy plane, then a rotation about x axis by 2

(b) A reflection in the yz plane, then a rotation about xy plane

(c) A rotation about y axis by 2

, then a reflection in the xz plane

(d) A rotation about x axis by 2

then a rotation about z axis by

2

B

D

AC y

x

z

BD

A

C

y

x

z




Q2. A particle moving in one dimension, is placed in an asymmetric square well potential V x as

sketched below:

The probability density p x in the ground state will most closely resemble

(a) (b)

(c) (d)

Q3. In a one-dimensional system, the boundary condition that the derivative of the wave function

x should be continuous at every point is applicable whenever

(a) There are bound or scattering states with definite momentum

(b) There is abound state and the potential is piecewise continuous.

(c) The wave function x is itself continuous everywhere

(d) There is a bounded state and the potential has no singularity anywhere

a V x a x

0 a ax

p

0 a ax

p

0 a ax

p

0 a ax

p




Q4. A British coin has a portrait of Queen Elizabeth II on the heads side and ‘ONE POUND’ written

on the tails side, while an Indian coin has a portrait of Mahatma Gandhi on the heads side and 10

RUPEES written on the tails side (see below)

These two coins are tossed simultaneously twice in succession.

The result of the first toss was ‘heads’ for both the coins. The probability that the result of the

second toss had a 10 RUPEES side is

(a)1

2 (b)

4

7 (c)

2

3 (d)

3

5

Q5. The sketch shown below illustrates the apparatus and results for a famous experiment. The graph

on the right is a polar plot of the number of electrons received in the detector.

(a) Electrons have spin half

(b) Electrons in a beam can behave as waves

(c) There are magnetic domains inside a nickel sample

(d) The energy levels of atoms in a metal are quantized

Q6. A beam of high energy neutrons is scattered from a metal lattice, where the spacing between

nuclei is around 0.4nm . In order to see quantum diffraction effects, the energy of the neutrons

must be around

(a) 7.85 MeV (b)5.11 meV (c) 78.5 meV (d)511 keV

00030

050

070

090

I

Ni

Inci

dent

bea

m

T

DGELECTRON GUN

DETECTOR




Q7. Drawing power from a 12V car battery a 9V stabilized DC voltage is required to power a car

stereo system, attached to the terminals A and B , as shown in the figure.

If a Zener diode with ratings 2 9V V and max 0.27P W , is connected as shown in the figure, for

the above purpose, the minimum series resistance SR must be

(a)111 (b)97 (c)100 (d)103

Q8. Consider the following linear model of a molecule of hydrogen cyanide HCN depicted below

It follows that the molar specific heat of hydrogen cyanide gas at constant pressure must be

(a) 4.5 R (b)5 R (c) 5.5 R (d) 6 R

Q9. A circular coil of conducting wire of radius a and n turns is placed in a uniform magnetic field

B

along the axis of the coil and is then made to undergo simple harmonic oscillations along the

direction of the axis. The current through the coil will be best described by

(a) (b)

(c) (d)

12V

SR

9zV V LRB

A

H C N

i t

0 t

i t

0 t

i t

0 t

i t

0 t




Q10. Two bodies A and B of equal mass are suspended from two rigid supports by separate massless

springs having spring constants 1k and 2k respectively. If the bodies oscillate

vertically such that their maximum velocities are equal the ratio of the

amplitude of oscillations of A to that of B is

(a) 2

1

k

k (b) 1

2

k

k (c) 1

2

k

k (d) 2

1

k

k

Q11. An ideal gas engine is run according to the cycle shown in the s T diagram below, where the

process from D to A is known to be isochoric (i.e. maintaining V constant)

the corresponding cycle in the p V diagram will most closely resemble

(a) (b)

(c) (d)

A B

2k1k

B

A

D

C

T

S

A

Dp

C

BV

A

D

p

C

B

V

A

D

p

C

B

VA

D

p C

B

V




Q12. Consider a thermal ensemble at temperature T , which is composed of identical quantum

harmonic oscillator of frequency 0 with non overlapping wave function. The probability that

there will been even number of energy quanta in the system is

(a) 0

1

exp / 1Bk T (b)

0

1

exp / 1Bk T

(c) 0

1

exp / 1Bk T (d) 0tan / 2 Bh k T

Q13. A point charge 0q is brought in front of a grounded conducting sphere. If the induced charge

density on the sphere is plotted such that the thickness of the black shading is proportional to the

charge, density the correct plot will most closely resemble

(a) (b)

(c) (d)

Q14. A set of polynomials of order n are given by the formula

2 2

1 exp exp2 2

nn

n n

x d xp x

dx

The polynomial 7p x of order 7n is

(a) 7 5 321 105 105 21x x x x

(b) 6 5 4 3 221 105 105 21x x x x x x

(c) 7 5 321 105 105x x x x

(d) 7 5 4 321 105 35 105x x x x x

q q

q q




Q15. A thermally insulated coffee mug contains 500 g of warm coffee at 080 C . Assuming that the

heat capacity of this liquid is 1 0 11cal g C and the latent heat of fusion for ice is 180cal g , the

amount of ice that must be dropped into the cup to convert it into cold coffee at 05 C is about

(a) 471g (b) 469 g (c) 421g (d) 441g

Q16. A particle of mass m , moving in one dimension, satisfies the modified Schrödinger equation

2 2

22

d d di u i

m dx dx dt

Where u is the velocity of the substrate? If now, this particle is treated as a Gaussian wave

packet peaked at wave number k , its group velocity will be gv

(a)hk

um

(b)hk

um

(c) hk

um

(d)2

hku

m

Q17. Consider the surface defined by 2 2 0ax by cz d , where , ,a b c and d are constants. If 1n

and 2n are unit normal vectors to the surface at the points , , 1,1,0x y z and 0,0,1

respectively and m is a unit vector normal to both 1n and 2n , then m

(a) 2 2

ai bj

a b

(b)

2 2 2

ˆˆ ˆai bj ck

a b c

(c)

2 2

ˆ ˆbi aj

a b

(d)

2 2 2

ˆˆ ˆ2 2

4 4

ai bj ck

a b c

Q18. The eigenvalues of a 3 3 matrix M are

1 2 32 1 1

and the eigenvectors are

1 2 3

1 1 11 1 11 2 0

e e e

The matrix M is

(a)1 1 01 0 10 1 1

(b)1 0 01 0 10 1 1

(c)0 1 11 0 01 0 2

(d) 1 0 10 1 11 1 0




Q19. The circuit shown below uses only NAND gates

The final output at C is

(a) A OR B (b) A XOR B

(c) A NOR B (d) A AND B

Q20. A plane electromagnetic wave travelling through vacuum has electric field E

and magnetic field

B

defined as

0ˆ ˆ expE i j E i t k x

0

ˆˆ ˆ expB i j k B i t k x

where 0E and 0B are real constants. The time-averaged pointing vector will be given by

(a) 200

0

ˆˆ ˆ 26

S E i j k

(b) 0 0

0 0

2 ˆˆ ˆ 2S E B i j k

(c) 200

0

31 ˆˆ ˆ 22

S E i j k

(d) 200

0

1 ˆˆ ˆ 22

S B i j k

Q21. Consider three straight coplanar, parallel wires of infinite length where the distance between

adjacent where is d . Each wire carries a current I in the same direction. The perpendicular

distance from the middle wire (on either side) where the magnetic field vanishes is

(a) 2

3

d (b)

3

d (c)

3

d (d)

2

3

d

BA

C




Q22. A particle of mass m is bounced on the ground with a velocity u making an angle of with the

ground. The coefficient of restitution for collisions between the particle and the ground is and

frictional effects are negligible both on the ground and in the air. The horizontal distance

travelled by the particle from the point of initial impact till it begins to slide along the ground is

(a) 2 1

sin1

u

g

(b)2

sin 22 1

u

g

(c)2

tan 21

u

g

(d)2

sin 21

u

g

Q23. A particle of mass m is placed on an inclined plane making an adjustable angle with the

horizontal, as shown in the figure. The coefficient of friction between the particle and the

inclined plane is .

If the inclined plane is moving horizontally with a uniform acceleration g

a

(see figure), the

value of for which the particle will remain at rest on the plane is

(a) 1tang a

g a

(b) 1tang a

g a

(c) 1cota g

a g

(d) 1cota g

a g

Q24. The three-dimensional object sketch on the right is made by taking a solid sphere

of uniform density (shaded) cavity (unshaded) as shown, which has diameter R

(a) 262

35MR (b) 231

20MR (c) 23

16MR (d) 231

70MR

u

a

m




Q25. The signal shown on the left side of the figure below is fed into the circuit shown on the right

side.

If the signal has time period s and the circuit has a natural frequency RC , then in the case when

s RC , the steady-state output will resemble

(a) (b)

(c) (d)

SECTION B

Q1. The electric charge density r inside a heavy spherical nucleus as a function of distance r

from the centre may be approximated most closely by

(a) (b)

(c) (d)

mV

0

mVinV R

C

inV

A

B

mV0

2 mV0mV

mV

0mV

mV

2 mV

0

0 r

r

0r

r

0r

r

0r

r




Q2. An excited gas, consisting of spinless charged particles, is confined in an infinite square well

potential of width a , is found to radiate a spectrum whose line (largest wavelength) has

wavelength 816 nm . If the width a of the well is halved to 2

a, the wavelength of the line

(fourth-largest wavelength) will be

(a) 1.224 m (b) 26.112 m (c) 0.306 m (d)1.088 g m

Q3. The molar equation of state of a gas at temperature T , pressure P and volume V is given by

2

RT ap

V b TV

where a and b are two constants and R is the gas constant. The critical temperature and

pressure for the gas will be

(a) 4

,27 4

cc c

RTaT P

Rb b (b)

8,

3 8c

c c

RTaT P

Rb b

(c)8

,27 8

cc c

RTaT P

Rb b (d) ,

27c

c c

RTaT P

Rb b

Q4. A pendulum is created by hanging a heavy bob of mass m from a rigid support (taken as zero

level of potential) symmetrically using two massless inextensible strings each of length l making

an equilateral triangles as shown in the figure below

A correct Lagrangian for the angular oscillations of the bob in the plane perpendicular to the

paper is

(a) 2 2 2 23sec 3 sin

8 2L ml mgl mgl

(b) 2 2 23 4 sin2

L ml mgl

(c) 2 2 2334 sin

8 2L ml mgl

(d) 2 233 sin

4 2

mgL m

l

l

l

lm




Q5. In a futuristic experiment, two rocket ships, each containing one astronaut Rakesh and Sunita

respectively, blasted off from the Earth’s surface simultaneously and travelled into space in

straight lines in opposite directions at uniform speeds of 0.3c and 0.5c respectively. They both

travelled in straight lines for some time then reversed direction smoothly and returned along the

same paths with the same speeds. It was found that they returned simultaneously after exactly

9.0 years. Which of the following a statement is correct?

(a) The ages of both Rakesh and Sunita increased by 9.0 years

(b) The ages of both Rakesh and Sunita increased by 3.06 years

(c) The age of Rakesh increased by about 8.6 years and that of Sunita by about 7.8 years

(d)The age of Rakesh increased by about 9.4 years and that of Sunita by about 10.4 years.

Q6. Consider a hydrogenic atom in its ground state ass conceived in Bohr’s

theory, where an electron of charge e is rotating about a central nucleus

of charge Ze in a circular orbit of radius 2 204 /a Ze in . In this

model, the magnetic field at a distance r from the nucleus, perpendicular

to the orbit will be

(a)1/ 22 2

02 2

0

11

4

Ze r

a a

(b)3/ 22 2

02 2

0

11

4

Ze r

a a

(c)3/ 22

02

0

1Ze r

a

(d)1/ 22 2

02 2

0

11

16

Ze r

a a

Q7. An array T has elements ijklT where , , , 1, 2,3,4i j k l . It is given that

ijkl ijkl ijlk klijT T T T

for all values of , , ,i j k l . The number of independent components in this array is

(a) 55 (b) 256 (c) 45 (d)1

Q8. The differential equation expdy

x xy xdx

, where 2y e at 1x , has the solution y

(a) 2exp x x (b) 1 exp exp 1x x x

(c) exp 1 1 lnx x (d) exp ln exp 1x x x

a

r




Q9. The wave function of a non-relativistic particle of mass m in a one-dimensional potential V x

has the form

a xx ae

Where x denotes the absolute value of the coordinate x . The potential is V x

(a) 0 (b) 2 1

2

aa x

m

(c) 2 x

m a

(d)

2ax

m

Q10. A dielectric interface is formed by two homogeneous and isotropic dielectrics 1 and 2 with

dielectric constants 4

3 and 1 respectively and it carries no residual free charge. A linearly

polarized electromagnetic wave is incident on the interface from dielectric1 at a point where the

unit normal to the surface is

1 ˆˆ 32

n i k

Pointing into the dielectric1. The incident wave, which is incident from 1 into 2 . Just before it

reaches the interface, has electric vector

11 0 expE iE i t z

c

where 0E is a real constant. The electric vector just after it crosses the interface is rotated from

1E

by an angle

(a) 1 4csc 3

19 (b) 1 1

sin2 19

(c) 1 2tan

5 3 (d)

6

Q11. A perfectly straight tunnel is dug between any two points on the surface of the Earth, which can

be treated as static sphere of uniform density . The tunnel does not necessarily pass through the

centre of the Earth. If a particle is allowed to slide without friction in this tunnel under the action

of the Earth’s gravity. It will execute simple harmonic motion with time period

(a) 3

G

(b)3

2 G (c)

3

2 G (d)

6

G




Q12. Te diffraction pattern due to a double slit experiment with two identical slits is recorded in infra-

red laser radiation of wavelength 1.2 m on a specially prepared photographic plate at a distance

of 1.8m from the centre of the slits. A photograph of the observed diffraction pattern is given

below.

The distance between the slits can be calculated as

(a) 4.32mm (b)5.04mm (c)5.76 mm (d)9.36 mm

Q13. In an experiment to measure the Earth’s mean albedo (i.e. fraction of solar energy reflected back

into space), the solar constant (i.e., flux of solar energy incident upon the Earth) was measured

as 21.37 kWm . Assuming the Earth to behave as a perfect blackbody at a uniform surface

temperature of 018 C the albedo is about

(a) 0.18 (b) 0.06 (c) 0.46 (d) 0.30

Q14. In a sample of germanium, at a temperature 77 K optical excitation create an average density of

1210 conduction electrons per 3cm . At this temperature, the electron and hole nobilities are equal

and given by

4 2 1 40.5 10 cm s V

The value of the Einstein diffusion coefficient for the electrons and holes is

(a) 2 2 13.3 10 m s (b) 3 2 16.6 10 m s

(c) 3 2 13.3 10 m s (d) 3 2 11.65 10 m s




Q15. In an experiment performed to determine the width of a thin transparent film of refractive index

2.6 deposited uniformly on the upper surface of a thick transparent glass block of refractive

index1.5 , a monochromatic laser beam of wavelength 550 nm was shone on the film at a

variable angle with the normal to the surface. If the intensity of the reflected beam was

maximum whens, the thickness of the film may be inferred to be

(a)1.12 g m (b) 13.2 m (c) 0.48 m (d) 2.92 m

1.5





GS 2019 ADMISSION TEST

Instructions for candidates for Integrated Ph.D. Programme in Physics

PLEASE READ THESE INSTRUCTIONS CAREFULLY BEFORE YOU ATTEMPT

THE QUESTIONS

This test consists of TWO sections.

SECTION A comprises 25 questions, numbered Q1 - Q25

These are questions on basic topics.

SECTION B comprises 15 questions, numbered Q1 - Q15

These may require somewhat more thought/knowledge.

ALL questions are Multiple-Choice Type. In each case, ONLY ONE option is correct.

Answer them by clicking the radio button next to the relevant option.

If your calculated answer does not match any of the given options exactly, you may mark the

closest one if it is reasonably close.

The grading scheme will be as follows:

Section A : +3 marks if correct; –1 mark if incorrect; 0 marks if not attempted

Section B : +5 marks if correct; 0 marks if incorrect or not attempted,

i.e. NO negative marks.

The invigilators will supply you with paper sheets for rough work.

Do NOT ask the invigilators for clarifications regarding the questions. They have been

instructed not to respond to any such queries. In case a correction/clarification is deemed

necessary, it will be announced in the examination hall.

You can get a list of useful physical constants on the reverse of this page. Make sure to use

only these values in answering the questions, especially where the options are numerical.




SECTION A

Q1. Consider the surface defined by 2 2 0ax by cz d , where , ,a b c and d are constants. If 1n

and 2n are unit normal vectors to the surface at the points , , 1,1,0x y z and 0,0,1

respectively and m is a unit vector normal to both 1n and 2n , then m

(a) 2 2

ai bj

a b

(b)

2 2

ˆ ˆbi aj

a b

(c)

2 2 2

ˆˆ ˆ2 2

4 4

ai bj ck

a b c

(d)

2 2 2

ˆˆ ˆai bj ck

a b c

Q2. The eigenvalues of a 3 3 matrix M are

1 2 32 1 1

and the eigenvectors are

1 2 3

1 1 11 1 11 2 0

e e e

The matrix M is

(a) 1 0 10 1 11 1 0

(b) 0 1 11 0 01 0 2

(c) 1 0 01 0 10 1 1

(d) 1 1 01 0 10 1 1

Q3. Which of the following operations will transform a tetrahedron ABCD with vertices as listed

below

x y z

A 0 0 0

B 1 0 0

C 0 1 0

D 0 0 2

B

D

AC y

x

z




Into a tetrahedron ABCD with vertices as listed below

x y z

A 0 0 0

B 0 1 0

C 0 0 1

D 2 0 0

Up to suitable translation?

(a) A rotation about x axis by 2

then a rotation about z axis by

2

(b) A reflection in the xy plane, then a rotation about x axis by 2

(c) A reflection in the yz plane, then a rotation about xy plane

(d) A rotation about y axis by 2

, then a reflection in the xz plane

Q4. A British coin has a portrait of Queen Elizabeth II on the ‘heads’ side and ‘ONE POUND’

written on the tails side, while an Indian coin has a portrait of Mahatma Gandhi on the heads side

and ‘10 RUPEES’ written on the ‘tails’ side (see below).

These two coins are tossed simultaneously twice in succession.

The result of the first toss was ‘heads’ for both the coins. The probability that the result of the

second toss had a ‘10 RUPEES’ side is

(a) 1

2 (b)

4

7 (c)

3

5 (d)

2

3

BD

A

C

y

x

z




Q5. A set of polynomials of order n are given by the formula

2 2

1 exp exp2 2

nn

n n

x d xp x

dx

The polynomial 7p x of order 7n is

(a) 7 5 4 321 105 35 105x x x x x (b) 6 5 4 3 221 105 105 21x x x x x x

(c) 7 5 321 105 105 21x x x x (d) 7 5 321 105 105x x x x

Q6. A particle of mass m is placed on an inclined plane making an adjustable angle with the

horizontal, as shown in the figure. The coefficient of friction between the particle and the

inclined plane is .

If the inclined plane is moving horizontally with a uniform acceleration g

a

(see figure), the

value of for which the particle will remain at rest on the plane is

(a) 1tang a

g a

(b) 1cota g

a g

(c) 1tang a

g a

(d) 1cota g

a g

Q7. Two bodies A and B of equal mass are suspended from two rigid supports by separate massless

springs having spring constants 1k and 2k respectively. If the bodies oscillate

vertically such that their maximum velocities are equal the ratio of the

amplitude of oscillations of A to that of B is

(a) 2

1

k

k (b) 2

1

k

k (c) 1

2

k

k (d) 2

1

k

k

Q8. A particle of mass m is bounced on the ground with a velocity u making an angle of with the

ground. The coefficient of restitution for collisions between the particle and the ground is and

a

m

A B

2k1k




frictional effects are negligible both on the ground and in the air. The horizontal distance

travelled by the particle from the point of initial impact till it begins to slide along the ground is

(a) 2

sin 22 1

u

g

(b) 2 1

sin1

u

g

(c) 2

tan 21

u

g

(d) 2

sin 21

u

g

Q9. The three-dimensional object sketch on the right is made by taking a solid sphere of uniform

density (shaded) with radius R , and scooping out a spherical cavity

(unshaded) as shown, which has diameter R .

In this object has mass M , its moment of inertia about the tangential axis

passing through the point where the spheres tough (as shown in the figure) is

(a) 23

16MR (b) 262

35MR (c) 231

70MR (d) 231

20MR

Q10. Consider three straight coplanar, parallel wires of infinite length where the distance between

adjacent where is d . Each wire carries a current I in the same direction. The perpendicular

distance from the middle wire (on either side) where the magnetic field vanishes is

(a) 3

d (b)

2

3

d (c)

3

d (d)

2

3

d

Q11. A point charge 0q is brought in front of a grounded conducting sphere. If the induced charge

density on the sphere is plotted such that the thickness of the black shading is proportional to the

charge, density the correct plot will most closely resemble

(a) (b)

(c) (d)

u

q

q

q

q




Q12. A circular coil of conducting wire of radius a and n turns, is placed in a uniform magnetic field

B

along the axis of the coil and is then made to undergo simple harmonic oscillations along the

direction of the axis. The current through the coil will be best described by

(a) (b)

(c) (d)

Q13. A plane electromagnetic wave travelling through vacuum has electric field E

and magnetic field

B

defined as

0ˆ ˆ expE i j E i t k x

0

ˆˆ ˆ expB i j k B i t k x

where 0E and 0B are real constants. The time-averaged pointing vector will be given by

(a) 0 0

0 0

2 ˆˆ ˆ 2S E B i j k

(b) 20

00

31 ˆˆ ˆ 22

S E i j k

(c) 200

0

ˆˆ ˆ 26

S E i j k

(d) 200

0

1 ˆˆ ˆ 22

S B i j k

Q14. A thermally insulated coffee mug contains 500 g of warm coffee at 080 C . Assuming that the

heat capacity of this liquid is 1 0 11cal g C and the latent heat of fusion for ice is 180cal g , the

amount of ice that must be dropped into the cup to convert it into cold coffee at 05 C is about

(a) 421g (b) 441g (c) 469 g (d) 471g

i t

0 t

i t

0 t

i t

0 t

i t

0 t




Q15. An ideal gas engine is run according to the cycle shown in the s T diagram below, where the

process from D to A is known to be isochoric (i.e. maintaining V constant).

the corresponding cycle in the p V diagram will most closely resemble

(a) (b)

(c) (d)

Q16. Consider a thermal ensemble at temperature T , which is composed of identical quantum

harmonic oscillator of frequency 0 with non-overlapping wave function. The probability that

there will be an even number of energy quanta in the system is

(a) 0

1

exp / 1Bk T (b)

0

1

exp / 1Bk T

(c) 0

1

exp / 1Bk T (d) 0tan / 2 Bh k T

B

A

D

C

T

S

A

Dp

C

BV

A

D

p

C

B

V

A

D

p

C

B

V

A

D

p C

B

V




Q17. Consider the following linear model of a molecule of hydrogen cyanide HCN depicted below.

It follows that the molar specific heat of hydrogen cyanide gas at constant pressure must be

(a) 6 R (b) 4.5 R (c) 5 R (d) 5.5 R

Q18. A beam of high energy neutrons is scattered from a metal lattice, where the spacing between

nuclei is around 0.4nm . In order to see quantum diffraction effects, the energy of the neutrons

must be around

(a) 7.85 MeV (b) 5.11 meV (c) 511 keV (d) 78.5 meV

Q19. A particle of mass m , moving in one dimension, satisfies the modified Schrödinger equation

2 2

22

d d di u i

m dx dx dt

Where u is the velocity of the substrate? If, now, this particle is treated as a Gaussian wave

packet peaked at wave number k , its group velocity will be gv

(a) 2

hku

m (b)

hku

m (c)

hku

m (d)

hku

m

Q20. In a one-dimensional system, the boundary condition that the derivative of the wavefunction

x should be continuous at every point is applicable whenever

(a) The wavefunction x is itself continuous everywhere.

(b) There is a bound state and the potential is piecewise continuous.

(c) There is a bounded state and the potential has no singularity anywhere.

(d) There are bound or scattering states with definite momentum.

H C N




Q21. A particle moving in one dimension, is placed in an asymmetric square well potential V x as

sketched below.

The probability density p x in the ground state will most closely resemble

(a) (b)

(c) (d)

Q22. The sketch shown below illustrates the apparatus and results for a famous experiment. The graph

on the right is a polar plot of the number of electrons received in the detector.

a V x a x

0 a ax

p

0 a ax

p

0 a ax

p

0 a ax

p

00030

050

070

090

I

Ni

Inci

dent

bea

m

T

DGELECTRON GUN

DETECTOR




(a) The energy levels of atoms in a metal are quantized.

(b) Electrons in a beam can behave as waves.

(c) Electrons have spin half.

(d) There are magnetic domains inside a nickel sample.

Q23. The signal shown on the left side of the figure below is fed into the circuit shown on the right

side.

If the signal has time period s and the circuit has a natural frequency RC , then in the case when

s RC , the steady-state output will resemble

(a) (b)

(c) (d)

Q24. Drawing power from a 12V car battery a 9V stabilized DC voltage is required to power a car

stereo system, attached to the terminals A and B , as shown in the figure.

If a Zener diode with ratings 2 9V V and max 0.27P W , is connected as shown in the figure, for

the above purpose, the minimum series resistance SR must be

(a)111 (b) 103 (c)100 (d) 97

12V

SR

9zV V LRB

A

mV

0

mVinV R

C

inV

A

B

mV0

2 mV

0mV

mV

0mV

mV

2 mV

0




Q25. The circuit shown below uses only NAND gates

The final output at C is

(a) A AND B (b) A OR B (c) A XOR B (d) A NOR B

SECTION B

Q1. The integral

0

sinxI dx e x

where x denotes the Dirac delta function, is

(a) 1 (b) exp

exp 1

(c) exp

exp 1

(d) 1

exp 1

Q2. Consider the complex function

, , ,f x y u x y iv x y

where 2 2, 2 2 3u x y x x y x

2 2, 3v x y y x x y

and is real. If it is known that ,f x y is analytic in the complex plane of z iy , then it can

be written

(a) 2 2f z z (b) 22f z z

(c) 2 22f zz z z (d) 2 3f z z

BA

C




Q3. A bead of mass of mass m slides under the influence of gravity along the frictionless interior of

a hemispherical cup of radius a sunk vertically into the ground with

its open side level with the ground (see sketch on the right). In terms

of spherical polar coordinates , set up with the centre of the

upper circle as origin, the Hamiltonian H for this system will be

(a) 2 2 2 2 2 2csc 2 sin2 2

mH a a mga

(b) 2 2 2 2 2 2sin 2 sin2 2

mH a a mga

(c) 2 2 2 22

1sin 2 sin

2 2H p p mga

ma

(d) 2 2 2 22

1csc 2 sin

2 2H p p mga

ma

Q4. On a compact stellar object the gravity is so strong that a body failing from rest will soon acquire

a velocity comparable with that of light. If the force on this body is F mg where m is the

relativistic mass and g is a constant, the velocity of this failing body will vary with time as

(a) 2

1

cv

cgt

(b) tanhgt

v cc

(c) 2

tanc gt

vc

(d) 1 expgt

v cc

Q5. A monsoon cloud has a flat bottom of surface area 2125km . It floats horizontally over the

ground at a level such that the base of the cloud is

1.13km above the ground (see figure). Due to friction

with the air below, the base of the cloud acquires a

uniform electric charge density. This keeps increasing

slowly with time.

When the uniform electric field below the cloud reaches

the value 12.4 MV m a lightning discharge occurs, and

1.13km




the entire charge of the cloud passes to the Earth below-which, in this case, behaves like a

grounded conductor.

Neglecting edge effects and inhomogeneities inside the cloud and the air below, the energy

released in this lighting discharge can be estimated, in kilowatt-hours (kWh) as about

(a) 910 (b) 510 (c) 10 (d) 110

Q6. The magnetic vector potential corresponding to a uniform magnetic field B

is often taken as

1

2A B x

This choice is

(a) valid in the Lorenz gauge (b) valid in the Coulomb gauge

(c) valid in the Weyl gauge (d) gauge invariant

Q7. An ideal monatomic gas at chemical potential 1eV and a temperature given by

0.1Bk T eV is in equilibrium with an adsorbing metal surface, i.e., there are isolated sites

distributed randomly on the metal surface where the gas atom can get bound. Each such binding

site can adsorb 0.1 or 2 atoms with the released energy being 0, 1eV and 1.9eV

respectively. The average number of adsorbed molecules at each site would be

(a) 1

1

e

e

(b) 1

12

e

e

(c)

2

1 2

e

e

(d) 1 2

1

e

e

Q8. Consider N non-interacting distinguishable particles in equilibrium at an absolute temperature

T . Each particle can only occupy one of two possible states of energy 0 and respectively

0 . The entropy of the system, in terms of Bk T

is

(a) ln 11B

eNk e

e

(b) ln 11B

eNk e

e

(c) ln 11B

eNk e

e

(d) ln 11B

eNk e

e




Q9. An electron in a hydrogen atom is in a state described by the wavefunction:

100 210 211 21, 1

1 2 2 1

5 510 10r x x x x

where nlm x denotes a normalized wavefunction of the hydrogen atom with the principal

quantum number n , angular quantum number and magnetic quantum number m .

Neglecting the spin-orbit intersection, the expectation values of ˆzL and 2L for this state are

(a) 23 9

,10 5

(b)

23 9,

4 25

(c)

23 9,

5 10

(d)

28 3,

10 5

Q10. A system of two spin - 1/ 2 particles 1 and 2 has the Hamiltonian

0 1 2ˆ ˆH

where

1

2 0ˆ0 1

2

0 1ˆ1 0

and 0 is a constant with the dimension of energy. The ground state of this system has energy

(a) 02 (b) 0 (c) 02 (d) 04

Q11. At time 0t , the wavefunctionof a particle in a harmonic oscillator potential of natural

frequency is given by

0 1 2

10 3 2 2 2 2

5

where n x denotes the eigenfunction belonging to the n -th eigenvalue of energy. At time

t , the wavefunction is found to be

0 1 23 2 2 2 25

i

the minimum value of is

(a) 2

(b) 2

(c) 2

3

(d)




Q12. At low temperature, the measured specific heat VC of a solid sample is found to depend on

temperature as 3/ 2 3VC aT bT

Where a and b are constants. This material has

(a) one fermionic excitation with dispersion relation 4k another bosonic excitation with

dispersion relation k ;

(b) one fermionic excitation with dispersion relation 2k another bosonic excitation with

dispersion relation 4k ;

(c) one bosonic excitation with dispersion relation 2k another bosonic excitation with


(d) one fermionic excitation with dispersion relation 2k another bosonic excitation with


Q13. Consider the following circuit

It is given that 100fC pF , and for 50inl nA .D C ,, out 1 .V V D C ,. Therefore, the bandwidth

of the above circuit is

(a) 15.8 Hz (b) 79.6 Hz (c) 145.3Hz (d) 200.4 Hz

inIoutV

fC

fR




Q14. The semi-empirical mass formula for a heavy nucleon ,Z A can be written, to some

approximation, as

2

2 2 2 2/3 52 3 41/ 2 1/ 2

1 2, p n

Z Z A ZM Z A c ZM c A Z M c A

A A A

where 2 2938 , 939p nM c MeV M c MeV , and 1 2 3 416, 18, 0.7, 23

all in MeV , where

5

+12 for even - even nuclei

-12 for odd - odd nuclei

0 for other

MeV

MeV

now, consider a spontaneous fission reaction

238 146 91 192 56 36 0U Ba Kr n

the energy released in this reaction will be close to

(a) 17.92keV (b) 19.2 MeV (c) 170 MeV (d) 190 MeV

Q15. The table below gives the properties of four unstable particles 0 0, , ,n

Mass

Particle 2/MeV c spin Principal decay mode

Muon 105.66 1

2 ee v v

Pion 139.57 0 v

Neutron 0n 939.56 1

2 0

en p e v

Lambda hyperon 0 1,115.68 1

2 0 p

If arranged in order of DECREASING decay lifetime, the above list will read

(a) 0 0, , ,n (b) 0 0, , ,n

(c) 0 0, , ,n (d) 0 0, , ,n

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