TIFR GS 2022 Question Paper Physics

GS2022 - Physics
Section A
(for both Integrated M.Sc.--Ph.D. and Ph.D. candidates)

This Section consists of 25 questions. All are of multiple-choice type. Mark only one option
on the online interface provided to you. If more than one option is marked, it will be
assumed that the question has not been attempted. A correct answer will get +3 marks, an
incorrect answer will get ̶ 1 mark.

1. Consider a square which can undergo rotations and reflections about its centre, where
making no transformation at all is counted as a rotation by 00 . The total number of
such distinct rotations and reflections which will keep the square unchanged is

✓ (a) 8 (b) 4

(c) 16 (d) 32

2. Consider the two-dimensional polar integral
2
𝑃 = ∫ 𝑑𝑟𝑑𝜃 𝑟 19 𝑒 −𝑟 sin8 𝜃 cos11 𝜃

If the integration is over only the first quadrant (0 ≤ 𝜃 ≤ 𝜋/2), the value of 𝑃 is

✓ (a) 180 (b) 88𝜋

(c) 20160 (d) 16𝜋

3. Consider a set of three 3-dimensional vectors
1 0 1
𝐴 = (0) 𝐵 = (1) 𝐶 = (1)
0 0 2
These vectors undergo a linear transformation
𝐴 → 𝐴′ = 𝕄𝐴 𝐵 → 𝐵 ′ = 𝕄𝐵 𝐶 → 𝐶 ′ = 𝕄𝐶
where 𝕄 is given by
1 1 4
𝕄 = (1 0 1 )
2 1 1
The volume of a parallelopiped whose sides are given by the transformed vectors 𝐴′ , 𝐵 ′
and 𝐶 ′ is

✓ (a) 8 (b) 4

(c) 2 (d) 16

4. A solid homogeneous sphere floats in water with a
portion sticking out above the water, as shown in the 𝑅/2
figure below. The height of the highest point above
the water surface is 𝑅/2 where 𝑅 is the radius of the 𝑅
sphere.
If the density of water is 1 g cm−3 , the density of the
material (in g cm−3) must be

✓ (a) 27/32 (b) 5/32

(c) 13/18 (d) 5/18

5. A particle of mass 𝑚 moves under the action of a central potential
𝑒2
𝑉(𝑟) = −
𝑟
where 𝑒 is a constant. Two vectors which remain conserved during the motion are
⃗ = 𝑟×𝑝
(i) the angular momentum 𝐿
⃗ =𝑝×𝐿
(ii) the Runge-Lenz vector 𝐾 ⃗ − 𝑚𝑒 2 𝑟̂ (where 𝑟̂ = 𝑟/𝑟)

The conserved energy 𝐸 of the particle can be written as

𝐾 2 − 𝑚2 𝑒 4 𝑚2 𝑒 4 − 𝐾 2
✓ (a) (b)
2𝑚𝐿2 2𝑚𝐿2

2𝑚𝐿2 2𝑚𝐿2
(c) (d)
𝐾 2 − 𝑚2 𝑒 4 𝑚2 𝑒 4 − 𝐾 2

6. The Principle of Linear Superposition of electron states in quantum mechanics is nicely
illustrated by the

✓ (a) Davisson-Germer experiment

(b) Compton scattering experiment

(c) Franck-Hertz experiment

(d) Millikan oil-drop experiment

7. A particle moves in one dimension 𝑥 under the influence of a potential 𝑉(𝑥) as
sketched in the figure below. The shaded region corresponds to infinite 𝑉, i.e., the
particle is not allowed to penetrate there.
𝑉(𝑥)

−𝑉0

If there is an energy eigenvalue 𝐸 = 0, then 𝑎 and 𝑉0 are related by

2
(𝑛 + 12) 𝜋 2 𝑛2 𝜋 2
✓ (a) 2
𝑎 𝑉0 = (b) 2
𝑎 𝑉0 =
2𝑚 2𝑚

(𝑛 + 12) 𝜋 2 𝑛 𝜋2
(c) 𝑎2 𝑉0 = (d) 𝑎2 𝑉0 =
2𝑚 2𝑚

8. In a hydrogenic atom of atomic number 𝑍, the probability amplitude that the nucleus
will capture an electron from its own K-shell is proportional to the overlap between
the nuclear wave-function
1
𝜓𝑛 (𝑟) = 𝑒 𝑟/𝑟𝑁
3
√8𝜋𝑟𝑁
and the electron wave-function
𝑍 3⁄2
𝜓𝑒 (𝑟) = 𝑒 −𝑍𝑟/𝑎0
3
√8𝜋𝑎0
where 𝑎0 is the Bohr radius and 𝑟𝑁 is the nuclear radius, which is known to vary as
𝑟𝑁 ∝ 𝑍 0.37 . The probability of electron capture, to a very good approximation, will be
proportional to 𝑍 𝛼 where 𝛼 is

✓ (a) 4.11 (b) 2.22

(c) 2.05 (d) 1.11

9. Treat the hydrogen molecule H2 as a rigid rotator. The next-to-largest wavelength in
its rotational spectrum is about 111 𝜇m. From this it can be estimated that the
separation between the pair of hydrogen atoms is about

✓ (a) 0.12 nm (b) 24.4 nm

(c) 61.4 nm (d) 3.07 μm

10. A falling raindrop, spherical in shape, with a diameter of 1 μm, acquires a uniform
negative charge due to friction with air. The electric field at a distance of 10 μm from
the surface of the droplet is measured to be 101 V m−1.
The number of excess electrons acquired by the droplet is

✓ (a) 7 (b) 7.02 × 106

(c) 1.4 × 1023 (d) 1414

11. An electromagnet is made by winding 𝑁 turns of wire around a wooden cylinder of
diameter 𝑑 and passing a current 𝐼 through it. When the current flows, a magnetic field
of magnitude 𝐵 is produced at a perpendicular distance 𝑧0 from the axis of the cylinder,
where 𝑧0 ≫ 𝑑.
If the number of turns 𝑁, the diameter of the wooden cylinder 𝑑 and the current 𝐼 are
all doubled, then the magnitude of the magnetic field will be 𝐵/2 at a distance 𝑧 =

✓ (a) 3.2 𝑧0 (b) 0.5 𝑧0

(c) 4.8 𝑧0 (d) 2.4 𝑧0

12. If an electron is set into oscillatory motion by the electric field of a laser of intensity
150 W m−2 and wavelength 554 nm, the amplitudes of its displacement and velocity,
respectively, are expected to be

5.1 × 10−18 m, 3.4 × 10−17 m,
✓ (a)
1.7 × 10−2 m s−1
(b)
1.0 × 10−1 m s −1

3.4 × 10−16 m, 3.4 × 10−18 m,
(c) (d)
1.7 × 10−1 m s−1 1.7 × 10−2 m s −1

13. A bicycle tyre is pumped with air to an internal pressure of 6 atm at 20o C, at which
point it suddenly bursts. Assuming the external pressure to be 1 atmosphere and the
subsequent sudden expansion to be adiabatic, the temperature immediately after the
burst is approximately

✓ (a) −97.5o C (b) −108.5o C

(c) 45.5o C (d) 216.0o C

14. A vertical cylinder of height 𝐻 is filled with an ideal gas of classical point particles
each of mass 𝑚 and is allowed to come to equilibrium under gravity at a temperature
𝑇. The mean height of these particles is

𝑘𝐵 𝑇 𝑚𝑔𝐻/𝑘𝐵 𝑇 𝐻 𝑚𝑔𝐻/𝑘𝐵 𝑇
✓ (a)
𝑚𝑔
(1 − 𝑚𝑔𝐻/𝑘 𝑇
𝑒 𝐵 −1
) (b)
3 𝑒 𝑚𝑔𝐻/𝑘𝐵 𝑇 + 1

𝑘𝐵 𝑇 2𝑚𝑔𝐻/𝑘𝐵 𝑇 𝐻 𝑚𝑔𝐻/𝑘𝐵 𝑇
(c) (1 − 𝑚𝑔𝐻/𝑘 𝑇 ) (d)
𝑚𝑔 𝑒 𝐵 +1 3 𝑒 𝑚𝑔𝐻/𝑘𝐵 𝑇 − 1

15. Two students A and B, measure the time period of a simple pendulum in the laboratory
using the same stopwatch but following two different methods.
― Student A measures the time taken for one oscillation and repeats it for 𝑁A number
of times and finds the average.
― Student B, on the other hand, measures the time taken for 𝑁B number of oscillations
and then computes the period.
Given that 𝑁A , 𝑁B ≫ 1, to ensure that both students measure the time period with the
same uncertainty, the relation between 𝑁A and 𝑁B must be

✓ (a) 𝑁A = 𝑁B2 (b) 𝑁A = √𝑁B

(c) 𝑁A = 𝑁B (d) ln 2 𝑁A = 𝑁B

16. A commercial advertisement for a solar power converter claims that when the
temperature of the plate (area 1.6 m2) absorbing 20% of the solar energy (solar constant
is about 1.36 kW m−2 s −1 ) reaches 1270 C and the rest of the device is at room
temperature (270 C), the system will deliver a power of 100 W.
If a prospective customer comes to you for advice about buying this device, your
advice should be that

✓ (a) it is an efficient device for the given specifications.

(b) the power delivered is very small for the given specifications.

(c) the advertisement is false and the device cannot deliver so much power.

(d) other similar devices are available which can deliver 1.5 – 2.0 times the power
with the same specifications.

17. Since the refractive index of water is 4/3, the angular velocity (in degrees per hour)
of the Sun at noon is perceived by a fish in the ocean deep below the surface as around

✓ (a) 11.3 (b) 15.0

(c) 13.2 (d) 20.0

18. The circuit diagram on the right shows a block A
representing a cubic structure comprising 12
identical resistances of 120 Ω each, whose body
diagonal vertices are connected to the rest of the
circuit with an inductor L= 10 mH, a resistor R=
100 Ω, and a capacitor C= 1 μF.

Now, the switch S is turned on at t = 0 . The
earliest time at which the current reaches a steady
value 𝐼0 is

✓ (a) zero (b) 100 𝜇s

(c) 200 𝜇s (d) infinite

19. Consider a circuit with an operational amplifier (op amp) and four resistors as sketched
below.

The output voltage 𝑉𝑂 is

✓ (a) −9 V (b) 0V

(c) −12 V (d) −6 V

20. It is required to design a circuit with an impedance Z(𝜔) such that
Z(𝜔) = 𝑖𝑘(𝜔 − 𝜔0 )
for a range of frequencies 𝜔 such that
|𝜔 − 𝜔0 |/𝜔0 ≪ 1
where 𝑘 and 𝜔0 are constant real numbers.
A possible design for this circuit would correspond to

✓ (a) (b)

(c) (d)

21. On a wet monsoon day at 12 noon, a thin film of oil of thickness 0.3 μm is formed on
a wet road. If the refractive index of oil and water are 1.475 and 1.333, respectively,
which of the following wavelengths of light will be reflected with maximum intensity?

✓ (a) 590 nm (b) 407 nm

(c) 443 nm (d) 640 nm

22. A gas of atoms, each of mass 𝑚, in thermal equilibrium at a temperature 𝑇, is radiating
with a frequency 𝜈0 . The Doppler broadening (full width at half maximum, or FWHM)
of the observed spectral line would be given by

2𝜈0 2 ln 2 𝑘𝐵 𝑇 𝜈0 2𝑘𝐵 𝑇
✓ (a) √ (b) √
𝑐 𝑚 𝑐 𝑚

2𝜈0 ln 2 𝑘𝐵 𝑇 2𝜈0 2𝑘𝐵 𝑇
(c) √ (d) √
𝑐 𝑚 𝑐 𝑚

23. Two particles, as specified in the table below, both enter a region of uniform magnetic
field in a direction perpendicular to the field direction.
Particle Rest Mass Kinetic Energy
Alpha 3.7 GeV 11.2 GeV
Deuteron 1.9 GeV 20.0 MeV
If both the particles then follow circular trajectories in the magnetic field, the ratio of
their time periods for one full revolution must be

✓ (a) 4.0 (b) 3.0

(c) 2.0 (d) 1.0

24. Natural potassium contains a radioactive component of 40K that has two decay modes.
― In the first mode, 40K undergoes a 𝛽 decay to the ground state of 40Ca.
― In the second mode, 40K undergoes an electron capture to the excited state of 40Ar,
followed by a single 𝛾 transition to the ground state of 40Ar.
The amount of radioactive 40K in a natural potassium (atomic weight of 39.089)
sample is known to be 0.0118 percent. It is also known that in the decay of 40K, for
every 100 𝛽 particles emitted, there number of 𝛾-photons emitted is 12.
If the number of 𝛽-particles emitted per second by 1 kg of natural potassium is
2.7 × 104 , the mean lifetime of 40K in years is

✓ (a) 1.9 × 109 (b) 1.3 × 109

(c) 1.7 × 109 (d) 1.1 × 108

25. The free electron model of metals (Drude model) explains several physical properties,
but cannot be used to explain

✓ (a) positive value of Hall coefficient

(b) magnetic susceptibility of the metal

(c) electrical conductivity of the metal

(d) thermal conductivity of the metal

Section B
(only for Integrated M.Sc.-Ph.D. candidates)

This Section consists of 15 questions. All are of multiple-choice type. Mark only one option
on the online interface provided to you. If more than one option is marked, it will be
assumed that the question has not been attempted. A correct answer will get +5 marks, an
incorrect answer will get 0 mark.

26. The value of the integral
+𝜋/2

∫ 𝑑𝑥 cosh 𝑘𝑥 2 sin2 𝑥
−𝜋/2
in the large-𝑘 limit, will be

✓ (a) 1 𝑘π2/4
𝑒
(b) 𝜋2
cosh ( )
𝑘π 4

(c) 1 𝜋2 (d) 1 𝑘π2 /4
cosh ( ) 𝑒
𝑘 2 π2 4 2𝑘π

27. A hollow metal sphere filled with a thick, highly viscous oil is rotating about a vertical
axis with an initial angular velocity 𝜔0 . However, there is a small hole at the bottom
of this sphere, through which drops of oil are leaking out vertically at a steady rate. The
variation of the angular velocity 𝜔(𝑡) of the sphere with time 𝑡 is best represented
graphically by

✓ (a) (b)

(c) (d)

28. A pendulum which is suspended from the ceiling of a train has time period 𝑇0 when
the train is stationary. When the train moves with a small but steady speed 𝑣 around a
horizontal circular track of radius 𝑅, the time period of the pendulum will be

−1⁄2 −1⁄2
𝑣 2 𝑇02 𝑣 2 𝑇02
✓ (a) 𝑇0 (1 − 2 ) (b) 𝑇0 (1 + 2 )
4𝜋 𝑅 4𝜋 𝑅

1/4 1/4
𝑣4 𝑣4
(c) 𝑇0 (1 − 2 2 ) (d) 𝑇0 (1 + 2 2 )
𝑔 𝑅 𝑔 𝑅

29. A cricket ball, bowled by a fast bowler, rises from
the pitch at an angle of 300 with a speed of 72
km/hr, then moves straight ahead and, at a height
of 0.5 m, strikes the flat surface of the bat held
firmly at rest in a horizontal position (see figure).
As a result, the ball bounces off elastically,
providing a return catch straight back to the
bowler.

If the coefficient of restitution between the bat and the ball is 0.577, the acceleration
due to gravity is 10 m s −2 and air resistance can be neglected, the catch will carry,
before hitting the ground, to a distance of approximately

✓ (a) 19.5 m (b) 37.0 m

(c) 9.5 m (d) 21.0 m

30. In a matrix mechanics formulation, a spin-1 particle has angular momentum
components
0 1 −1
ℏ ℏ 2 0 0
𝐿𝑥 = ( 1 √2 0 ) 𝐿𝑧 = (0 −1 −1)
2 2
−1 0 −√2 0 −1 −1
It follows that 𝐿𝑦 =

0 −𝑖 𝑖 0 𝑖 −𝑖
ℏ ℏ
(𝑖 0 −𝑖√2) (−𝑖 0 𝑖√2)
✓ (a) 2 (b) 2
−𝑖 𝑖√2 0 𝑖 −𝑖√2 0

0 √2 0 0 −√2 0
(c) √2 ℏ (√2 1 0) (d) √2 ℏ (−√2 −1 0)
0 0 −1 0 0 1

31 Two equal positive point charges 𝑄 = +1
. are placed on either side of an 𝑥-axis
normal to a grounded infinite conducting
plane at distances of 𝑥 = +2 units and 𝑥 =
−1 unit respectively (see figure) w.r.t. the
point of intersection of the axis with the
conducting plane as origin.
The electrostatic potential along the axis
will correspond to the graph in

✓ (a) (b)

(c) (d)

32. Two co-axial solenoids A and B, one placed completely inside the other, have the
following parameters:
Solenoid No of turns Length Diameter
A 1000 50 cm 2 cm
B 2000 50 cm 4 cm
The mutual inductance between the solenoids is

✓ (a) 1.58 mH (b) 125.7 mH

(c) 395.0 mH (d) 12.57 mH

33. A particle of mass 𝑚 in a three-dimensional potential well has a Hamiltonian of the
form
𝑝𝑥2 𝑝𝑦2 𝑝𝑧2 1 1
𝐻= + + + 𝑚𝜔2 𝑥 2 + 𝑚𝜔2 𝑦 2 + 2𝑚𝜔2 𝑧 2
2𝑚 2𝑚 2𝑚 2 2

where 𝜔 is a constant. If there are two identical spin-½ particles in this potential having
a total energy
𝐸 = 6ℏ𝜔
the entropy of the system will be

✓ (a) 𝑘𝐵 ln 14 (b) 𝑘𝐵 ln 16

(c) 𝑘𝐵 ln 12 (d) 𝑘𝐵 ln 10

34. A quantum dot is constructed such that it has just three energy levels, with energies 𝐸,
2𝐸 and 3𝐸 respectively. The chemical potential in the system has the value 𝜇 = 2𝐸
and the temperature is given by
𝐸
𝑇=
2𝑘𝐵
The expected number of electrons populating the quantum dot will be

✓ (a) 3.0 (b) 2.5

(c) 1.5 (d) 4.0

35. The non-inverting amplifier shown in the
figure on the right is constructed using a non-
ideal operational amplified (op amp) with a
finite open loop gain 𝐴.
The value of feedback fraction is
𝑅2
𝐵= = 0.1
𝑅1 + 𝑅2
If the gain 𝐴 varies such that
104 < 𝐴 < 105
then the approximate percentage variation in
the closed loop gain will be.

✓ (a) 0.09 % (b) 0.0 %

(c) 0.9 % (d) 9.0 %

36. In a standardized entrance exam, the passing rates for the past 10 years are tabulated
below.
Year 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
Passing Rate 22 % 16 % 23 % 21% 22 % 14 % 17 % 20 % 24 % 21 %
If 1000 candidates appear for the exam every year, the probability that more than 250
students will pass the exam this year is about

✓ (a) 6% (b) 20%

(c) 25% (d) 0.1%

37. A spectrographic method to search for exoplanets is by measuring its velocity along
the line of sight, using the Doppler shift in the spectrum. If a star of mass 𝑀 and a
planet of mass 𝑚 are moving around their common centre of mass, this component of
velocity will vary periodically with an amplitude.
2𝜋𝐺𝑁 1/3 𝑚
𝐴=( )
𝑇 𝑀2/3
For a particular planet-star system, if the time period is 𝑇 = (12 ± 0.3) years, and
𝐴 and 𝑀 are measured with an accuracy of 3% each, then the error in the measurement
of the mass 𝑚 is

✓ (a) 3.7 % (b) 8.5 %

(c) 5.8 % (d) 6.3%

38. A satellite used to make Google Earth images carries on board a telescope which must
be designed, when operating at a wavelength 𝜆, to be able to resolve objects on the
ground of length as small as 𝛿.
If the satellite goes around the Earth in a circular orbit with uniform speed 𝑣, the
minimum diameter 𝐷min of the telescope mirror can be determined in terms of 𝑅, the
radius of the Earth, and 𝑔, the acceleration due to gravity at the surface, to be

1.22𝜆 𝑔𝑅 2 1.22𝜆 𝑔𝑅 2 𝑅
✓ (a)
𝛿
( 2 − 𝑅)
𝑣
(b) (1 + )
𝛿 𝑣2 𝜆

1.22𝜆 𝑔𝑅 2 1.22𝜆 𝑔𝑅 3
(c) (d) √
𝛿 𝜆𝑣 2 𝛿 𝑣2

39. An observer O, moving with relativistic speed 𝑣 away from a fixed plane mirror M in
a line perpendicular to the mirror surface, sends a pulse of light of wavelength λ
towards the mirror.

𝑣
O

M
The wavelength of the light reflected back to the observer will be

𝑐+𝑣 𝑐 + 2𝑣
✓ (a) λ(
𝑐−𝑣
) (b) λ(
𝑐 − 2𝑣
)

𝑐−𝑣 𝑐 − 2𝑣
(c) λ√ (d) λ ( )
𝑐+𝑣 𝑐 + 2𝑣

40. The low-temperature specific heat of a certain material is primarily due to acoustic
phonons. The frequency 𝜔 of a phonon is related to its wavevector 𝑘 by 𝜔 = 𝑐𝑘,
where 𝑐 is the speed of sound in the material. The phonons have a Bose distribution
1
𝑛(𝑘) =
𝑒 ℏ𝑐𝑘/𝑘𝐵 𝑇 − 1
and the energy of a phonon has a maximum possible value 𝜔𝐷 .
In a two-dimensional sample, the specific heat at low-temperatures behaves as

𝑇 2 𝑇 3
✓ (a) ( ) (b) ( )
𝜔𝐷 𝜔𝐷

𝑇 3/2 𝑇
(c) ( ) (d)
𝜔𝐷 𝜔𝐷

Section C
(only for Ph.D. candidates)

This Section consists of 15 questions. All are of multiple-choice type. Mark only one option
on the online interface provided to you. If more than one option is marked, it will be
assumed that the question has not been attempted. A correct answer will get +5 marks, an
incorrect answer will get 0 mark.

26. Consider the inner product in the space of normalisable functions defined on the
interval [−1,1]
1
⟨𝑓|𝑔⟩ = ∫ 𝑑𝑥 (1 + 𝑥 2 ) 𝑓(𝑥) 𝑔(𝑥)
−1

The projection of the vector 1 along the vector 𝑥 2 is

14 2 16 35 2
✓ (a) 𝑥 (b) √ 𝑥
9 15 24

16 2 35 2
(c) 𝑥 (d) √ 𝑥
15 24

27. A dumbbell consists of two small spherical
masses 𝑀 each, connected by a thin massless rod
of length 2ℓ.
This dumbbell is centred at the origin, and is
rotating about the 𝑧-axis with a uniform angular
velocity 𝜔, making an angle 𝜃 with the 𝑧-axis (see
figure).
Neglecting effects due to gravity, at the instant
when the dumbbell is wholly in the 𝑦𝑧-plane (as
shown in the figure), the magnitude of torque
about the origin will be

✓ (a) 𝑀ℓ2 𝜔2 sin 2𝜃 (b) 2𝑀ℓ2 𝜔2 sin2 𝜃

(c) 2𝑀ℓ2 𝜔2 cos 2 𝜃 (d) zero

28. A system with two generalized coordinates (𝑞1 , 𝑞2 ) is described by the Lagrangian
3 3
𝐿 = 𝑚 (𝑞̇ 12 + 2𝑞̇ 1 𝑞̇ 2 + 𝑞̇ 22 ) − 𝑘 ( 𝑞12 + 2𝑞1 𝑞2 + 𝑞22 )
2 2
where 𝑚 is the mass, and 𝑘 is a constant.
This system can execute oscillations with two possible time periods

2𝑚 𝑚
✓ (a) 𝑇 = 2𝜋√ and 𝑇 = 2𝜋√
𝑘 2𝑘

𝑚 𝑚
(b) 𝑇 = 2𝜋√ (5 − 2√6) and 𝑇 = 2𝜋√ (5 + 2√6)
2𝑘 2𝑘

𝑚 𝑚
(c) 𝑇 = 𝜋√ (1 − √15) and 𝑇 = 𝜋√ (1 + √15)
𝑘 𝑘

2𝑚 3𝑚
(d) 𝑇 = 2𝜋√ and 𝑇 = 2𝜋√
3𝑘 2𝑘

29. A system was formed of three spin-½ particles A, B and C, respectively and it was
prepared in an initial state
|𝜓⟩ = 𝑐1 ∣↑↑↑⟩ + 𝑐2 ∣↑↑↓⟩ + 𝑐3 ∣↑↓↑⟩ + 𝑐4 ∣↑↓↓ ⟩ + 𝑐5 ∣↓↑↑⟩ + 𝑐6 ∣↓↑↓⟩ + 𝑐7 ∣↓↓↑⟩ + 𝑐8 ∣↓↓↓⟩
where the symbols ∣↑⟩ and ∣↓⟩ indicate states with 𝑆𝑧 = +½ (spin-up) and 𝑆𝑧 = −½
(spin-down) respectively.
A measurement was made on the system in the initial state and this identified the spin
state of the particle A to be ∣↓⟩ (spin-down). Now the expectation value of 〈𝑆𝑧 〉 for the
particle C could be calculated as

✓ (a) |𝑐5 |2 + |𝑐7 |2 − |𝑐6 |2 −|𝑐8 |2
|𝑐5 |2 + |𝑐7 |2 + |𝑐6 |2 + |𝑐8 |2

(b) 𝑐5 + 𝑐7 − 𝑐6 − 𝑐8
|𝑐5 | + |𝑐7 |2 + |𝑐6 |2 + |𝑐8 |2
2

(c) (𝑐5∗ + 𝑐7∗ − 𝑐6∗ − 𝑐8∗ )(𝑐5 + 𝑐7 − 𝑐6 − 𝑐8 )
|𝑐5 |2 + |𝑐7 |2 + |𝑐6 |2 + |𝑐8 |2

(d) (𝑐5 + 𝑐7 )∗ (𝑐5 + 𝑐7 ) − (𝑐6 + 𝑐8 )∗ (𝑐6 + 𝑐8 )
|𝑐5 |2 + |𝑐7 |2 + |𝑐6 |2 + |𝑐8 |2

30. A particle is confined to a one-dimensional lattice with a lattice spacing δ. In the
position space, the Hamiltonian operator for this particle is given by the matrix
⋱
... ... 0 0 0 0
. . . 2 −1 0 0 0
0 −1 2 −1 0 0
ℋ = 𝐸0
0 0 −1 2 −1 0
0 0 0 −1 2 . . .
0 0 0 0 ... ...
( ⋱)
Noting that it commutes with the generator 𝑇 of translations
⋱
... ... 0 0 0 0
... 0 1 0 0 0
0 0 0 1 0 0
𝒯=
0 0 0 0 1 0
0 0 0 0 0 ...
0 0 0 0 ... ...
( ⋱)
where 𝑇 = 𝑒 𝑖𝒫 δ/ℏ in terms of the momentum operator 𝒫, the energy of a state with
momentum 𝑝 will be

✓ (a) 4𝐸0 sin2(𝑝δ⁄2ℏ) (b) 𝐸0 cos (𝑝𝛿 ⁄ℏ)

(c) 𝐸0 sin (𝑝𝛿 ⁄ℏ) (d) 𝐸0 (𝑝𝛿 ⁄2ℏ)2

31. According to a standard table, the refractive index of water at 4o C is 1.33 at a
wavelength of 590 nm. However, a carefully performed experiment in the lab yielded
a refractive index of 1.41.
Which one of the following statements could be the explanation of this discrepancy?

✓ (a) The experiment was performed at a wavelength lower than 590 nm.

(b) The experiment was performed at a wavelength higher than 590 nm.

(c) The water sample was at a temperature lower than 4o C.

(d) The water sample was at a temperature much higher than 4o C.

32. The power radiated by a point charge 𝑞 moving rapidly with a uniform speed 𝑣 in a
circle of radius 𝑅 will be

2 2
𝑞2𝑐 𝑣2 𝑞4𝑐 2 𝑣2
✓ (a) ( ) (b) ( )
6𝜋𝜀0 𝑅 2 𝑐 2 − 𝑣 2 6𝜋𝜀0 𝑅 2 𝑐 2 − 𝑣 2

𝑞3𝑐 𝑣2 𝑞2𝑐 3 𝑣2
(c) (d)
6𝜋𝜀0 𝑅 4 𝑐 2 − 𝑣 2 6𝜋𝜀0 𝑅 3 𝑐 2 − 𝑣 2

33. A pseudo-potential 𝑉12 between every pair of particles in an ideal gas is to be
constructed which will reproduce the effects of quantum statistics if the gas particles
are bosonic in nature. A correct formula for this, in terms of the inter-particle distance
𝑟12 and a mean distance 𝜆, will be of the form

✓
2 2
(a) 𝑉12 = −𝑘𝐵 𝑇 ln (1 + 𝑒 −2𝜋𝑟12 /𝜆 )

2 2
(b) 𝑉12 = −𝑘𝐵 𝑇 ln (1 − 𝑒 −2𝜋𝑟12 /𝜆 )

2 2
(c) 𝑉12 = +𝑘𝐵 𝑇 ln (1 + 𝑒 −2𝜋𝑟12 /𝜆 )

2 2
(d) 𝑉12 = +𝑘𝐵 𝑇 ln (1 − 𝑒 −2𝜋𝑟12 /𝜆 )

34. Three students A, B and C are given identical counters and each is asked to measure
the number of gamma rays emitted per second by a given radioactive source. They are
expected to perform the counting many times and find the mean and the standard
deviation. The students find the following:
Student A B C
Measurement (counts/second) 482 ±22 495 ±10 501 ±22
If a counting experiment conducted previously by the instructor on this same sample
with another identical counter had recorded exactly 30,000 gamma rays in a minute,
then which of the following interpretations is valid?

✓ (a) The measurement by student B is too precise to be believable.

(b) The measurement of student B is more correct than that of student A.

(c) The measurements of A and C have too large standard deviations.

(d) The measurement of C is much more precise than that of A.

35. A thyratron consists of a tube filled with
Xenon gas which can be used as a high-
power electrical switch. Electrons are emitted
from a cathode K heated by a filament F, and
made to accelerate to some energy 𝐸 by a
voltage 𝑉 applied across the anode plate P.
Electrons that scatter from the Xe atoms get
deviated from their path and hit the shield S,
which is a conducting envelope that
transports the electrons back to ground
potential (see figure on the right). The rest of
the electrons strike the plate and contribute to
the plate current 𝑖𝑃 .
Which of the following graphs of the variation of the plate current 𝑖𝑃 with increase in
the accelerating voltage 𝑉 could indicate the wave nature of the electron ?

✓ (a) (b)

(c) (d)

36. In a semiclassical approach, the Hamiltonian of a He atom is modified by adding a
magnetic interaction term between the two electrons, of the form

ℋ𝐼 = 𝐴2 𝑆1 ⋅ 𝑆2

where 𝑆1 and 𝑆2 are the electron spins and 𝐴2 is a coupling constant. This leads, for
the configuration 1𝑠 2 , to the energy shift

✓ (a) − 3𝐴2 ⁄4 (b) +3𝐴2 ⁄4

(c) +𝐴2 ⁄4 (d) − 𝐴2 ⁄4

37. In the shell model of the nucleus, it is known that orbitals get filled in the order
1𝑠1/2 1𝑝3/2 1𝑝1/2 1𝑑5/2 2𝑠1/2 1𝑑3/2 and so on

For a nucleus of 188O the two neutrons outside the doubly-magic core of 168O will
occupy the same orbital. The allowed value of 𝙹𝑝 will be

✓ (a) 4+ (b) 5+

(c) 2− (d) 3+

38. From the knowledge that you already have about the length of one year and the fact
that the Sun subtends 0.5o in the sky, the average density of the Sun can be computed
in kg-m-3 as

✓ (a) 1.7 × 103 (b) 7.5 × 103

(c) 1.7 × 102 (d) 7.5 × 102

39. At very low temperatures, the electrical resistivity of most metals is dominated by

✓ (a) collisions of conduction electrons with impurity atoms and lattice vacancies.

(b) absorption of conduction electrons by ions in the lattice.

(c) collisions of conduction electrons with lattice phonons.

(d) transfer of conduction electrons to the valence band.

40. There are two conceivable channels by which a vector 𝜌0 meson can decay into a pair
of pseudoscalar pions. These are
𝜌0 → 𝜋 0 + 𝜋 0 and 𝜌0 → 𝜋 + + 𝜋 −
The probability that the decay takes place through the process 𝜌0 → 𝜋 + + 𝜋 − is
approximately

✓ (a) 1 (b) 𝑚𝜋0 /2𝑚𝜋+

(c) 𝑚𝜋2 + /𝑚𝜌2 (d) zero