GATE 2023 Question Paper PH Physics

The Graduate Aptitude Test in Engineering

GATE 2023
Question Paper
GATE is an entrance examination conducted in India that
primarily tests the comprehensive understanding of
undergraduate subjects in engineering and sciences for
admission into technical postgraduate programs.

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General Aptitude (GA)
Q.1 – Q.5 Carry ONE mark Each

Q.1 “You are delaying the completion of the task. Send _______ contributions at the
earliest.”

(A) you are

(B) your

(C) you’re

(D) yore

Q.2 References : ______ : : Guidelines : Implement

(By word meaning)

(A) Sight

(B) Site

(C) Cite

(D) Plagiarise

Page 1 of 48

Q.3 In the given figure, PQRS is a parallelogram with PS = 7 cm, PT = 4 cm and
PV = 5 cm. What is the length of RS in cm? (The diagram is representative.)

7
P S

5 90°
4
90° V
Q T R

(A) 20
7

(B) 28
5

(C) 9
2

(D) 35
4

Page 2 of 48

Q.4 In 2022, June Huh was awarded the Fields medal, which is the highest prize in
Mathematics.

When he was younger, he was also a poet. He did not win any medals in the
International Mathematics Olympiads. He dropped out of college.

Based only on the above information, which one of the following statements can be
logically inferred with certainty?

(A) Every Fields medalist has won a medal in an International Mathematics Olympiad.

(B) Everyone who has dropped out of college has won the Fields medal.

(C) All Fields medalists are part-time poets.

(D) Some Fields medalists have dropped out of college.

Page 3 of 48

Q.5 A line of symmetry is defined as a line that divides a figure into two parts in a way
such that each part is a mirror image of the other part about that line.

The given figure consists of 16 unit squares arranged as shown. In addition to the
three black squares, what is the minimum number of squares that must be coloured
black, such that both PQ and MN form lines of symmetry? (The figure is
representative)

P N

M
Q

(A) 3

(B) 4

(C) 5

(D) 6

Page 4 of 48

Q.6 – Q.10 Carry TWO marks Each

Q.6 Human beings are one among many creatures that inhabit an imagined world. In
this imagined world, some creatures are cruel. If in this imagined world, it is given
that the statement “Some human beings are not cruel creatures” is FALSE, then
which of the following set of statement(s) can be logically inferred with certainty?

(i) All human beings are cruel creatures.
(ii) Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.

(A) only (i)

(B) only (iii) and (iv)

(C) only (i) and (ii)

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

Q.7 To construct a wall, sand and cement are mixed in the ratio of 3:1. The cost of sand
and that of cement are in the ratio of 1:2.

If the total cost of sand and cement to construct the wall is 1000 rupees, then what
is the cost (in rupees) of cement used?

(A) 400

(B) 600

(C) 800

(D) 200

Page 5 of 48

The World Bank has declared that it does not plan to offer new financing to Sri
Q.8 Lanka, which is battling its worst economic crisis in decades, until the country has
an adequate macroeconomic policy framework in place. In a statement, the World
Bank said Sri Lanka needed to adopt structural reforms that focus on economic
stabilisation and tackle the root causes of its crisis. The latter has starved it of
foreign exchange and led to shortages of food, fuel, and medicines. The bank is
repurposing resources under existing loans to help alleviate shortages of essential
items such as medicine, cooking gas, fertiliser, meals for children, and cash for
vulnerable households.

Based only on the above passage, which one of the following statements can be
inferred with certainty?

(A) According to the World Bank, the root cause of Sri Lanka’s economic crisis is that
it does not have enough foreign exchange.

(B) The World Bank has stated that it will advise the Sri Lankan government about how
to tackle the root causes of its economic crisis.

(C) According to the World Bank, Sri Lanka does not yet have an adequate
macroeconomic policy framework.

(D) The World Bank has stated that it will provide Sri Lanka with additional funds for
essentials such as food, fuel, and medicines.

Q.9 The coefficient of 𝑥 4 in the polynomial (𝑥 − 1)3 (𝑥 − 2)3 is equal to _______.

(A) 33

(B) ̶ 3

(C) 30

(D) 21

Page 6 of 48

Q.10 Which one of the following shapes can be used to tile (completely cover by
repeating) a flat plane, extending to infinity in all directions, without leaving any
empty spaces in between them? The copies of the shape used to tile are identical
and are not allowed to overlap.

(A) circle

(B) regular octagon

(C) regular pentagon

(D) rhombus

Page 7 of 48

Q.11 – Q.25 Carry ONE mark Each

Q.11 Which one of the following entropy (S) - temperature (T) diagrams CORRECTLY
represents the Carnot cycle 𝑎𝑏𝑐𝑑𝑎 shown in the 𝑷-𝑽 diagram?

(A)

(B)

(C)

(D)

Page 8 of 48

Q.12 Which one of the following is a dimensionless constant?

(A) Permittivity of free space

(B) Permeability of free space

(C) Bohr magneton

(D) Fine structure constant

Page 9 of 48

Q.13 Choose the most appropriate matching of the items in Column 1 with those in
Column 2.

Column 1 Column 2

(i) PIN diode P. Voltage regulation

(ii) Tunnel diode Q. Radio frequency and
microwave devices

(iii) Zener diode R. Optoelectronic detection

(iv) Photo diode S. Oscillator

(A) (i) - Q; (ii) - S; (iii) - P; (iv) - R

(B) (i) - R; (ii) - Q; (iii) - P; (iv) - S

(C) (i) - R; (ii) - S; (iii) - P; (iv) - Q

(D) (i) - P; (ii) - Q; (iii) - R; (iv) - S

Page 10 of 48

Q.14 The atomic number of an atom is 6. What is the spectroscopic notation of its
ground state, according to Hund’s rules?

(A) 3
𝑃0

(B) 3
𝑃1

(C) 3
𝐷3

(D) 3
𝑆1

Q.15
𝐻 is the Hamiltonian, 𝐿⃗ the orbital angular momentum and 𝐿𝑧 is the
⃗ . The 1𝑠 state of the hydrogen atom in the non-relativistic
z-component of 𝐿
formalism is an eigen function of which one of the following sets of operators?

(A) 𝐻, 𝐿2 and 𝐿𝑧

(B)
𝐻, 𝐿⃗, 𝐿2 and 𝐿𝑧

(C) 𝐿2 and 𝐿𝑧 only

(D) 𝐻 and 𝐿𝑧 only

Page 11 of 48

Q.16 The Hall experiment is carried out with a non-magnetic semiconductor. The
current 𝐼 is along the 𝑥 -axis and the magnetic field 𝐵 is along the 𝑧-axis. Which
one of the following is the CORRECT representation of the variation of the
magnitude of the Hall resistivity 𝜌𝑥𝑦 as a function of the magnetic field?

(A)

(B)

(C)

(D)

Page 12 of 48

Q.17 Consider a two dimensional Cartesian coordinate system in which a rank 2
0 1
contravariant tensor is represented by the matrix ( ). The coordinate system
1 0
is rotated anticlockwise by an acute angle 𝜃 with the origin fixed. Which one of
the following matrices represents the tensor in the new coordinate system?

(A) 0 cos 2𝜃
( )
− sin 2𝜃 0

(B) sin 2𝜃 cos 2𝜃
( )
cos 2𝜃 − sin 2𝜃

(C) sin 2𝜃 −cos 2𝜃
( )
cos 2𝜃 sin 2𝜃

(D) sin 2𝜃 0
( )
0 − cos 2𝜃

Q.18 A compound consists of three ions X, Y and Z. The Z ions are arranged in an FCC
1
arrangement. The X ions occupy of the tetrahedral voids and the Y ions occupy
6
1
of the octahedral voids. Which one of the following is the CORRECT chemical
3
formula of the compound?

(A) XY2Z4

(B) XYZ3

(C) XYZ2

(D) XYZ4

Page 13 of 48

Q.19 For a non-magnetic metal, which one of the following graphs best represents the
𝐶
behaviour of vs. 𝑇 2 , where 𝐶 is the heat capacity and 𝑇 is the temperature?
𝑇

(A)

(B)

(C)

(D)

Page 14 of 48

Q.20 For nonrelativistic electrons in a solid, different energy dispersion relations (with
effective masses 𝑚𝑎 ∗
, 𝑚𝑏∗ , and 𝑚𝑐∗ ) are schematically shown in the plots.
Which one of the following options is CORRECT?

(A) 𝑚𝑎∗ = 𝑚𝑏∗ = 𝑚𝑐∗

(B) 𝑚𝑏∗ > 𝑚𝑐∗ > 𝑚𝑎∗

(C) 𝑚𝑐∗ > 𝑚𝑏∗ > 𝑚𝑎∗

(D) 𝑚𝑎∗ > 𝑚𝑏∗ > 𝑚𝑐∗

Page 15 of 48

Q.21 The figure schematically shows the 𝑀 (magnetization) - 𝐻 (magnetic field) plots
for certain types of materials. Here 𝑀 and 𝐻 are plotted in the same scale and
units. Which one of the following is the most appropriate combination?

(A) (Q) - Paramagnet; (R) - Type-I Superconductor; (S) - Antiferromagnet

(B) (P) - Paramagnet; (Q) - Diamagnet; (R) - Type-I Superconductor

(C) (P) - Paramagnet; (Q) - Antiferromagnet; (R) - Type-I Superconductor

(D) (P) - Diamagnet; (R) - Paramagnet; (S) - Type-I Superconductor

Page 16 of 48

Q.22 Graphene is a two dimensional material, in which carbon atoms are arranged in a
honeycomb lattice with lattice constant 𝑎 . As shown in the figure, 𝑎1 and 𝑎2
are two lattice vectors. Which one of the following is the area of the first Brillouin
zone for this lattice?

(A) 8𝜋 2
3√3𝑎2

(B) 4𝜋 2
3√3𝑎2

(C) 8𝜋 2
√3𝑎2

(D) 4𝜋 2

√3𝑎2

Page 17 of 48

Q.23 A 60Co nucleus emits a 𝛽 -particle and is converted to 60Ni* with 𝐽 𝑃 = 4+ ,
which in turn decays to the 60Ni ground state with 𝐽 𝑃 = 0+ by emitting two
photons in succession, as shown in the figure. Which one of the following
statements is CORRECT?

(A) 4+ → 2+ is an electric octupole transition

(B) 4+ → 2+ is a magnetic quadrupole transition

(C) 2+ → 0+ is an electric quadrupole transition

(D) 2+ → 0+ is a magnetic quadrupole transition

Page 18 of 48

Q.24 Which one of the following options is CORRECT for the given logic circuit?

(A) P = 1, Q = 1; X = 0

(B) P = 1, Q = 0; X = 1

(C) P = 0, Q = 1; X = 0

(D) P = 0, Q = 0; X = 1

Q.25 An atom with non-zero magnetic moment has an angular momentum of
magnitude √12 ℏ. When a beam of such atoms is passed through a Stern-
Gerlach apparatus, how many beams does it split into?

(A) 3

(B) 7

(C) 9

(D) 25

Page 19 of 48

Q.26 A 4 × 4 matrix 𝑀 has the property 𝑀 † = −𝑀 and 𝑀 4 = 𝟏, where 𝟏 is the
4 × 4 identity matrix. Which one of the following is the CORRECT set of
eigenvalues of the matrix 𝑀?

(A) (1, 1, −1, −1)

(B) (𝑖, 𝑖, −𝑖, −𝑖)

(C) (𝑖, 𝑖, 𝑖, −𝑖)

(D) (1, 1, −𝑖, −𝑖)

Q.27 The Ξ 0∗ particle is a member of the Baryon decuplet with isospin state
1 1
| 𝐼, 𝐼3 〉 = | , 〉 and strangeness quantum number −2. In the quark model,
2 2

which one of the following is the flavour part of the Ξ 0∗ wavefunction?

(A) 1
(𝑢𝑠𝑠 − 𝑠𝑠𝑢)
√2

(B) 1
(𝑢𝑠𝑠 + 𝑠𝑢𝑠 + 𝑠𝑠𝑢)
√3

(C) 1
(𝑢𝑠𝑠 + 𝑠𝑠𝑢)
√2

(D) 1
(𝑢𝑠𝑠 − 𝑠𝑢𝑠 + 𝑠𝑠𝑢)
√3

Page 20 of 48

Q.28 Which of the following is(are) the CORRECT option(s) for the Joule-Thomson
effect?

(A) It is an isentropic process

(B) It is an isenthalpic process

(C) It can result in cooling as well as heating

(D) For an ideal gas it always results in cooling

Q.29 The deuteron is a bound state of a neutron and a proton. Which of the following
statements is(are) CORRECT?

(A) The deuteron has a finite value of electric quadrupole moment due to non-
spherical electronic charge distribution

(B) The magnetic moment of the deuteron is equal to the sum of the magnetic
moments of the neutron and the proton

(C) The deuteron state is an admixture of 3S1 and 3D1 states

(D) The deuteron state is an admixture of 3S1 and 3P1 states

Page 21 of 48

Q.30 The Geiger-Muller counter is a device to detect 𝛼, 𝛽 and 𝛾 radiations. It is a
cylindrical tube filled with monatomic gases like argon, and polyatomic gases
such as ethyl alcohol. The inner electrode is along the axis of the cylindrical tube
and the outer electrode is the tube. Which of the following statements is(are)
CORRECT?

(A) Argon is used so that ambient light coming from the surroundings do not produce
any signal in the detector

(B) Ethyl alcohol is used as a quenching gas

(C) The electric field strength decreases from the axis to the edge of the tube and the
direction of the field is radially outward

(D) The electric field increases from the axis to the edge of the tube and the field
direction is radially inward

Page 22 of 48

Q.31 Consider an isolated magnetized sphere of radius 𝑅 with a uniform magnetization
𝑀⃗⃗ along the positive 𝑧 direction, with the north and south poles of the sphere
lying on the 𝑧 axis. It is given that the magnetic field inside the sphere is
⃗ = 2𝜇0 𝑀
𝐵 ⃗⃗ , where 𝜇0 is the permeability of vacuum. Which of the following
3
statements is(are) CORRECT?

(A) The bound volume current density is zero

(B) The bound surface current density has maximum magnitude at the equator, where
⃗⃗ |
this magnitude equals |𝑀

(C) 2
⃗ = − 𝑀
The auxiliary field 𝐻 ⃗⃗
3

(D) Far from the sphere, the magnetic field is due to a dipole of moment 𝑚
⃗⃗ , where
𝑚⃗⃗⃗ 𝐵
= 𝑧̂
4𝜋𝑅 3 2𝜇0

Page 23 of 48

Q.32 Which of the following options represent(s) linearly independent pair(s) of
functions of a real variable 𝑥 ?

(A) 𝑒 𝑖𝑥 and 𝑒 −𝑖𝑥

(B) 𝑥 and 𝑒 𝑥

(C) 2𝑥 and 2−3+𝑥

(D) 𝑒 𝑖𝑥 and sin 𝑥

Page 24 of 48

Q.33 In the vector model of angular momentum applied to atoms, what is the minimum
angle in degrees (in integer) made by the orbital angular momentum vector and
the positive 𝑧 axis for a 2𝑝 electron?

Q.34 For a transistor amplifier, the frequency response is such that the mid band voltage
gain is 200. The cutoff frequencies are 20 Hz and 20 kHz. What is the ratio
(rounded off to two decimal places) of the voltage gain at 10 Hz to that at 100
kHz?

Q.35
An electric field as a function of radial coordinate 𝑟 has the form
𝑒 −𝑟2
𝐸⃗ = 𝛼 𝑟̂ , where 𝛼 is a constant. Assume that dimensions are
𝑟
appropriately taken care of. The electric flux through a sphere of radius √2,
Φ
centered at the origin, is Φ. What is the value of (rounded off to two decimal
2𝜋𝛼
places)?

Page 25 of 48

Q.36 – Q.65 Carry TWO marks Each

Q.36 It is given that the electronic ground state of a diatomic molecule 𝑋2 has even
parity and the nuclear spin of 𝑋 is 0. Which one of the following is the
CORRECT statement with regard to the rotational Raman spectrum (𝐽 is the
rotational quantum number) of this molecule?

(A) Lines of all 𝐽 values are present

(B) Lines have alternating intensity in the ratio of 3: 1

(C) Lines of only even 𝐽 values are present

(D) Lines of only odd 𝐽 values are present

Page 26 of 48

Q.37 An input voltage in the form of a square wave of frequency 1 kHz is given to a
circuit, which results in the output shown schematically below. Which one of the
following options is the CORRECT representation of the circuit?

(A)

(B)

(C)

(D)

Page 27 of 48

Q.38 A simple harmonic oscillator with an angular frequency 𝜔 is in thermal
2𝑘𝐵 𝑇
equilibrium with a reservoir at absolute temperature 𝑇, with 𝜔 = . Which
ℏ
one of the following is the partition function of the system?

(A) 𝑒
𝑒2 − 1

(B) 𝑒
𝑒2 + 1

(C) 𝑒
𝑒−1

(D) 𝑒
𝑒+1

Page 28 of 48

Q.39 Which one of the following options is the most appropriate match between the
items given in Column 1 and Column 2?

Column 1 Column 2

(i) Visible light P. Transition between core energy
levels of atoms

(ii) X-rays Q. Transition between nuclear
energy levels

(iii) Gamma rays R. Pair production

(iv) Thermal neutrons S. Crystal structure determination

T. Photoelectric effect

(A) (i) - T; (ii) - P,S,T; (iii) - Q,R; (iv) - S

(B) (i) - P,T; (ii) - S; (iii) - R,S; (iv) - S,T

(C) (i) - T; (ii) - R,S; (iii) - Q,R; (iv) - S

(D) (i) - S,T; (ii) - P,S; (iii) - R,T; (iv) - S

Page 29 of 48

Q.40 A rod 𝑃𝑄 of proper length 𝐿 lies along the 𝑥 -axis and moves towards the positive
3𝑐
𝑥 direction with speed 𝑣 = with respect to the ground (see figure), where
5
𝑐 is the speed of light in vacuum. An observer on the ground measures the
positions of 𝑃 and 𝑄 at different times 𝑡𝑃 and 𝑡𝑄 respectively in the ground
9𝐿
frame, and finds the difference betwen them to be . What is the value
10
of 𝑡𝑄 − 𝑡𝑃 ?

(A) 𝐿
3𝑐

(B) 𝐿
5𝑐

(C) 𝐿
6𝑐

(D) 2𝐿
3𝑐

Page 30 of 48

Q.41 2𝛼
A symmetric top has principal moments of inertia 𝐼1 = 𝐼2 = , 𝐼3 = 2𝛼
3
about a set of principal axes 1, 2, 3 respectively, passing through its center of
mass, where 𝛼 is a positive constant. There is no force acting on the body and
1
the angular speed of the body about the 3-axis is 𝜔3 = rad/s. With what
8
angular frequency in rad/s does the angular velocity vector 𝜔
⃗ 1 precess about the
3-axis?

(A) 2

(B) 3

(C) 5

(D) 7

Page 31 of 48

Q.42 A particle of mass m is free to move on a frictionless horizontal two dimensional
𝑘
(𝑟, 𝜃) plane, and is acted upon by a force 𝐹 = − 3 𝑟̂ with 𝑘 being a
2𝑟
positive constant. If 𝑝𝑟 and 𝑝𝜃 are the generalised momenta corresponding to
𝑑𝑝𝑟
𝑟 and 𝜃 respectively, then what is the value of ?
𝑑𝑡

(A) 𝑝𝜃2 − 2𝑚𝑘
2𝑚𝑟 3

(B) 2𝑝𝜃2 − 𝑚𝑘
𝑚𝑟 3

(C) 𝑝𝜃2 − 2𝑚𝑘
𝑚𝑟 3

(D) 2𝑝𝜃2 − 𝑚𝑘
2𝑚𝑟 3

Page 32 of 48

Q.43 Consider two real functions

𝑈(𝑥, 𝑦) = 𝑥𝑦(𝑥 2 − 𝑦 2 ) ,

𝑉(𝑥, 𝑦) = 𝑎𝑥 4 + 𝑏𝑦 4 + 𝑐𝑥 2 𝑦 2 + 𝑘 ,

where 𝑘 is a real constant and 𝑎, 𝑏, 𝑐 are real coefficients. If 𝑈(𝑥, 𝑦) + 𝑖 𝑉(𝑥, 𝑦)
is analytic, then what is the value of 𝑎 × 𝑏 × 𝑐 ?

(A) 1
8

(B) 3
28

(C) 5
36

(D) 3
32

Page 33 of 48

Q.44 Young’s double slit experiment is performed using a beam of C60 (fullerene)
molecules, each molecule being made up of 60 carbon atoms. When the slit
separation is 50 nm, fringes are formed on a screen kept at a distance of 1 m from
the slits. Now, the experiment is repeated with C70 molecules with a slit
separation of 92.5 nm. The kinetic energies of both the beams are the same. The
position of the 4th bright fringe for C60 will correspond to the 𝑛th bright fringe
for C70. What is the value of 𝑛 (rounded off to the nearest integer) ?

(A) 5

(B) 6

(C) 7

(D) 8

Q.45
A neutron beam with a wave vector 𝑘 ⃗ and an energy 20.4 meV diffracts from a
⃗ ′ . One of the diffraction peaks is observed
crystal with an outgoing wave vector 𝑘
−1
for the reciprocal lattice vector 𝐺 of magnitude 3.14 Å . What is the diffraction
⃗ makes with the plane?
angle in degrees (rounded off to the nearest integer) that 𝑘
−27
(Use mass of neutron = 1.67 × 10 Kg)

(A) 15

(B) 30

(C) 45

(D) 60

Page 34 of 48

Q.46 In the first Brillouin zone of a rectangular lattice (lattice constants 𝑎 = 6 Å and
𝑏 = 4 Å), three incoming phonons with the same wave vector
〈1.2 Å−1 , 0.6 Å−1 〉 interact to give one phonon. Which one of the following is
the CORRECT wave vector of the resulting phonon?

(A) 〈2.56 Å−1 , 0.23 Å−1 〉

(B) 〈3.60 Å−1 , 1.80 Å−1 〉

(C) 〈0.48 Å−1 , 0.23 Å−1 〉

(D) 〈3.60 Å−1 , −0.80 Å−1 〉

Page 35 of 48

Q.47 For a covalently bonded solid consisting of ions of mass 𝑚, the binding potential
can be assumed to be given by
𝑟
𝑟 −
𝑈(𝑟) = − 𝜖 ( ) 𝑒 𝑟0 ,
𝑟0

where 𝜖 and 𝑟0 are positive constants. What is the Einstein frequency of the
solid in Hz ?

(A)
1 𝜖𝑒
√
2𝜋 𝑚𝑟02

(B)
1 𝜖
√
2𝜋 𝑚𝑒𝑟02

(C)
1 2𝜖
√
2𝜋 𝑚𝑒𝑟02

(D)
1 𝜖𝑒
√
2𝜋 2𝑚𝑟02

Page 36 of 48

Q.48 In a hadronic interaction, 𝜋 0 ’s are produced with different momenta, and they
immediately decay into two photons with an opening angle 𝜃 between them.
Assuming that all these decays occur in one plane, which one of the following
figures depicts the behaviour of 𝜃 as a function of the 𝜋 0 momentum 𝑝?

(A)

(B)

(C)

(D)

Page 37 of 48

Q.49 A particle has wavefunction
2 2 2
𝜓(𝑥, 𝑦, 𝑧) = 𝑁 𝑧 𝑒 −𝛼(𝑥 +𝑦 +𝑧 ) ,
where 𝑁 is a normalization constant and 𝛼 is a positive constant. In this state,
which one of the following options represents the eigenvalues of 𝐿2 and 𝐿𝑧
respectively?
𝑚
Some values of 𝑌ℓ are :

1 3 3
𝑌00 = √ , 𝑌10 = √ cos 𝜃 , 𝑌1±1 = ∓√ sin 𝜃 𝑒 ±𝑖𝜙
4𝜋 4𝜋 8𝜋

(A) 0 and 0

(B) ℏ2 and −ℏ

(C) 2ℏ2 and 0

(D) ℏ2 and ℏ

Page 38 of 48

Q.50 The wavefunction of a particle in one dimension is given by

𝑀, −𝑎 < 𝑥 < 𝑎
𝜓(𝑥) = {
0, otherwise.
Here 𝑀 and 𝑎 are positive constants. If 𝜑(𝑝) is the corresponding momentum
space wavefunction, which one of the following plots best represents |𝜑(𝑝)|2 ?

(A)

(B)

(C)

(D)

Page 39 of 48

Q.51 Consider a particle in a two dimensional infinite square well potential of side 𝐿,
with 0 ≤ 𝑥 ≤ 𝐿 and 0 ≤ 𝑦 ≤ 𝐿. The wavefunction of the particle is zero only
𝐿
along the line 𝑦 = , apart from the boundaries of the well. If the energy of the
2
particle in this state is 𝐸 , what is the energy of the ground state?

(A) 1
𝐸
4

(B) 2
𝐸
5

(C) 3
𝐸
8

(D) 1
𝐸
2

Page 40 of 48

Q.52 1
Consider two non-identical spin particles labelled 1 and 2 in the spin product
2
1 1 1 1
state | , 〉 | , − 〉. The Hamiltonian of the system is
2 2 2 2
4λ
𝐻= 𝑆1 ⋅ 𝑆2 ,
ℏ2

where 𝑆1 and 𝑆2 are the spin operators of particles 1 and 2, respectively, and 𝜆
is a constant with appropriate dimensions. What is the expectation value of 𝐻 in
the above state?

(A) −𝜆

(B) −2𝜆

(C) 𝜆

(D) 2𝜆

Page 41 of 48

Q.53 1
A spin particle is in a spin up state along the 𝑥 -axis (with unit vector 𝑥̂ ) and
2
1 1
is denoted as | , 〉 . What is the probability of finding the particle to be in a
2 2 𝑥
spin up state along the direction 𝑥̂ ′ , which lies in the 𝑥𝑦-plane and makes an
angle 𝜃 with respect to the positive 𝑥 -axis, if such a measurement is made?

(A) 1 𝜃
cos2
2 4

(B) 𝜃
cos2
4

(C) 1 𝜃
cos2
2 2

(D) 𝜃
cos2
2

Page 42 of 48

Q.54 Different spectral lines of the Balmer series (transitions 𝑛 → 2, with 𝑛 being the
principal quantum number) fall one at a time on a Young’s double slit apparatus.
The separation between the slits is 𝑑 and the screen is placed at a constant distance
from the slits. What factor should 𝑑 be multiplied by to maintain a constant fringe
width for various lines, as 𝑛 takes different allowed values?

(A) 𝑛2 − 4
4𝑛2

(B) 𝑛2 + 4
4𝑛2

(C) 4𝑛2
𝑛2 − 4

(D) 4𝑛2
𝑛2 + 4

Page 43 of 48

Q.55 Under parity and time reversal transformations, which of the following statements
is(are) TRUE about the electric dipole moment 𝐩 and the magnetic dipole
moment 𝝁 ?

(A) 𝐩 is odd under parity and 𝝁 is odd under time reversal

(B) 𝐩 is odd under parity and 𝝁 is even under time reversal

(C) 𝐩 is even under parity and 𝝁 is odd under time reversal

(D) 𝐩 is even under parity and 𝝁 is even under time reversal

Q.56 Consider the complex function

𝑧 2 sin 𝑧
𝑓 (𝑧) = .
(𝑧 − 𝜋)4
At 𝑧 = 𝜋 , which of the following options is(are) CORRECT?

(A) The order of the pole is 4

(B) The order of the pole is 3

(C) 𝜋
The residue at the pole is
6

(D) 2𝜋
The residue at the pole is
3

Page 44 of 48

Q.57 Consider the vector field 𝑉 ⃗ consisting of the velocities of points on a thin
horizontal disc of radius 𝑅 = 2 m, moving anticlockwise with uniform angular
⃗ | , then
speed 𝜔 = 2 rad/sec about an axis passing through its center. If 𝑉 = |𝑉
̂ are
which of the following options is(are) CORRECT ? (In the options, 𝑟̂ and 𝜃
unit vectors corresponding to the plane polar coordinates 𝑟 and 𝜃 ).

You may use the fact that in cylindrical coordinates (𝑠, 𝜙, 𝑧) (s is the distance
from the 𝑧-axis), the gradient, divergence, curl and Laplacian operators are:

⃗∇𝑓 = 𝜕𝑓 𝑠̂ + 1 𝜕𝑓 𝜙̂ + 𝜕𝑓 𝑧̂ ;
𝜕𝑠 𝑠 𝜕𝜙 𝜕𝑧

⃗ . 𝐴 = 1 𝜕 (𝑠𝐴𝑠 ) + 1 𝜕𝐴𝜙 + 𝜕𝐴𝑧 ;
∇
𝑠 𝜕𝑠 𝑠 𝜕𝜙 𝜕𝑧

⃗ × 𝐴 = (1 𝜕𝐴𝑧 − 𝜕𝐴𝜙) 𝑠̂ + (𝜕𝐴𝑠 − 𝜕𝐴𝑧) 𝜙̂
∇
𝑠 𝜕𝜙 𝜕𝑧 𝜕𝑧 𝜕𝑠

1 𝜕 𝜕𝐴𝑠
+ ( (𝑠𝐴𝜙 ) − ) 𝑧̂ ;
𝑠 𝜕𝑠 𝜕𝜙
2 2
⃗∇2 𝑓 = 1 𝜕 (𝑠 𝜕𝑓) + 12 𝜕 𝑓2 + 𝜕 𝑓2 .
𝑠 𝜕𝑠 𝜕𝑠 𝑠 𝜕𝜙 𝜕𝑧

(A) ⃗∇𝑉 = 2𝑟̂

(B) ⃗ .𝑉
∇ ⃗ =2

(C) ⃗∇ × 𝑉
⃗ = 4𝑧̂ , where 𝑧̂ is a unit vector perpendicular to the (𝑟, 𝜃) plane

(D) ⃗∇2 𝑉 = 4 at 𝑟 = 1.5 m
3

Page 45 of 48

Q.58 A slow moving 𝜋 − particle is captured by a deuteron (𝑑) and this reaction
produces two neutrons (𝑛) in the final state, i.e., 𝜋 − + 𝑑 → 𝑛 + 𝑛. Neutron
and deuteron have even intrinsic parities, whereas 𝜋 − has odd intrinsic parity. 𝐿
and 𝑆 are the orbital and spin angular momenta, respectively of the system of two
neutrons. Which of the following statements regarding the final two-neutron state
is(are) CORRECT?

(A) It has odd parity

(B) 𝐿 + 𝑆 is odd

(C) 𝐿 = 1, 𝑆 = 1

(D) 𝐿 = 2, 𝑆 = 0

Page 46 of 48

Q.59 Two independent electrostatic configurations are shown in the figure.
Configuration (𝐼) consists of an isolated point charge 𝑞 = 1 C, and configuration
(𝐼𝐼) consists of another identical charge surrounded by a thick conducting shell of
inner radius 𝑅1 = 1 𝑚 and outer radius 𝑅2 = 2 𝑚, with the charge being at
𝜖0 𝜖
the center of the shell. 𝑊𝐼 = ∫ 𝐸𝐼2 𝑑𝑉 and 𝑊𝐼𝐼 = 0 ∫ 𝐸𝐼𝐼2 𝑑𝑉 ,
2 2
where 𝐸𝐼 and 𝐸𝐼𝐼 are the magnitudes of the electric fields for configurations (𝐼)
and (𝐼𝐼) respectively, 𝜖0 is the permittivity of vacuum, and the volume
8𝜋 1
integrations are carried out over all space. If |𝑊𝐼 − 𝑊𝐼𝐼 | = , what is
𝜖0 𝑛
the value of the integer 𝑛?

Q.60 In pion nucleon scattering, the pion and nucleon can combine to form a short lived
bound state called the Δ particle (𝜋 + 𝑁 → Δ). The masses of the pion,
nucleon and the Δ particle are 140 MeV/𝑐 2 , 938 MeV/𝑐 2 and 1230 MeV/𝑐 2 ,
respectively. In the lab frame, where the nucleon is at rest, what is the minimum
energy (in MeV/𝑐 2 , rounded off to one decimal place) of the pion to produce the
Δ particle?

Q.61 Consider an electromagnetic wave propagating in the 𝑧-direction in vacuum, with
⃗ =𝐵
the magnetic field given by 𝐵 ⃗ 0 𝑒 𝑖(𝑘𝑧−𝜔𝑡) . If 𝐵0 = 10−8 T, the average

power passing through a circle of radius 1.0 m placed in the 𝑥𝑦 plane is 𝑃 (in
𝐶2 103 𝑃
Watts). Using 𝜖0 = 10−11 , what is the value of (rounded off to
𝑁 𝑚2 𝜋
one decimal place)?

Page 47 of 48

Q.62 An 𝛼-particle is emitted from the decay of Americium (Am) at rest, i.e.,
241 237 241 237
94Am → 92U + 𝛼 . The rest masses of 94Am, 92U and 𝛼 are 224.544
GeV/c2, 220.811 GeV/c2 and 3.728 GeV/c2 respectively. What is the kinetic
energy (in MeV/𝑐 2 , rounded off to two decimal places) of the 𝛼-particle?

Q.63 1
Consider 6 identical, non-interacting, spinatoms arranged on a crystal lattice
2
at absolute temperature 𝑇. The 𝑧-component of the magnetic moment of each of
these atoms can be ± 𝜇𝐵 . If 𝑃 and 𝑄 are the probabilities of the net magnetic
𝑃
moment of the solid being 2𝜇𝐵 and 6𝜇𝐵 respectively, what is the value of (in
𝑄
integer)?

Q.64 Two identical, non-interacting 4He2 atoms are distributed among 4 different non-
degenerate energy levels. The probability that they occupy different energy levels
is 𝑝. Similarly, two 3He2 atoms are distributed among 4 different non-degenerate
energy levels, and the probability that they occupy different levels is 𝑞. What is
𝑝
the value of (rounded off to one decimal place)?
𝑞

Q.65 Two identical bodies kept at temperatures 800 K and 200 K act as the hot and the
cold reservoirs of an ideal heat engine, respectively. Assume that their heat
capacity (𝐶) in Joules/K is independent of temperature and that they do not
undergo any phase change. Then, the maximum work that can be obtained from
the heat engine is 𝑛 × 𝐶 Joules. What is the value of 𝑛 (in integer)?

END OF QUESTION PAPER

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