GATE 2024 Question Paper for Physics (PH)

GATE
2024
Question Paper | Answer Key
Graduate Aptitude Test in Engineering
(GATE) is a prestigious national-level exam
that assesses candidates for
comprehensive understanding in various
undergraduate-level subjects in
Engineering, Technology, Science,
Architecture, and Humanities.

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

Q.1 If ‘→’ denotes increasing order of intensity, then the meaning of the words

[smile → giggle → laugh] is analogous to [disapprove → ________ → chide].

Which one of the given options is appropriate to fill the blank?

(A) reprove

(B) praise

(C) reprise

(D) grieve

Q.2 Find the odd one out in the set: {19, 37, 21, 17, 23, 29, 31, 11}

(A) 21

(B) 29

(C) 37

(D) 23

Page 1 of 44

Q.3 In the following series, identify the number that needs to be changed to form the
Fibonacci series.

1, 1, 2, 3, 6, 8, 13, 21,…

(A) 8

(B) 21

(C) 6

(D) 13

Q.4 The real variables 𝑥, 𝑦, 𝑧, and the real constants 𝑝, 𝑞, 𝑟 satisfy
𝑥 𝑦 𝑧
2
= 2
=
𝑝𝑞 − 𝑟 𝑞𝑟 − 𝑝 𝑟𝑝 − 𝑞 2

Given that the denominators are non-zero, the value of 𝑝𝑥 + 𝑞𝑦 + 𝑟𝑧 is

(A) 0

(B) 1

(C) 𝑝𝑞𝑟

(D) 𝑝2 + 𝑞 2 + 𝑟 2

Page 2 of 44

Q.5 Take two long dice (rectangular parallelepiped), each having four rectangular faces
labelled as 2, 3, 5, and 7. If thrown, the long dice cannot land on the square faces
and has 1/4 probability of landing on any of the four rectangular faces. The label on
the top face of the dice is the score of the throw.

If thrown together, what is the probability of getting the sum of the two long dice
scores greater than 11?

(A) 3/8

(B) 1/8

(C) 1/16

(D) 3/16

Page 3 of 44

Q.6 – Q.10 Carry TWO marks Each

Q.6 In the given text, the blanks are numbered (i)−(iv). Select the best match for all the
blanks.

(i) (ii)
Prof. P merely a man who narrated funny stories. in his blackest
moments he was capable of self-deprecating humor.

(iii) (iv)
Prof. Q a man who hardly narrated funny stories. in his blackest
moments was he able to find humor.

(A) (i) was (ii) Only (iii) wasn’t (iv) Even

(B) (i) wasn’t (ii) Even (iii) was (iv) Only

(C) (i) was (ii) Even (iii) wasn’t (iv) Only

(D) (i) wasn’t (ii) Only (iii) was (iv) Even

Q.7 How many combinations of non-null sets A, B, C are possible from the subsets of
{2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?

(A) 28

(B) 27

(C) 18

(D) 19

Page 4 of 44

Q.8 The bar chart gives the batting averages of VK and RS for 11 calendar years from
2012 to 2022. Considering that 2015 and 2019 are world cup years, which one of
the following options is true?

140
120
Batting Average

100 VK RS
80
60
40
20
0
2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

Calendar Year

(A) RS has a higher yearly batting average than that of VK in every world cup year.

(B) VK has a higher yearly batting average than that of RS in every world cup year.

(C) VK’s yearly batting average is consistently higher than that of RS between the two
world cup years.

(D) RS’s yearly batting average is consistently higher than that of VK in the last three
years.

Page 5 of 44

Q.9 A planar rectangular paper has two V-shaped pieces attached as shown below.

This piece of paper is folded to make the following closed three-dimensional object.

The number of folds required to form the above object is

(A) 9

(B) 7

(C) 11

(D) 8

Page 6 of 44

Q.10 Four equilateral triangles are used to form a regular closed three-dimensional object
by joining along the edges. The angle between any two faces is

(A) 30°

(B) 60°

(C) 45°

(D) 90°

Page 7 of 44

Q.11 – Q.35 Carry ONE mark Each

Q.11 If 𝐹1 (𝑄, 𝑞) = 𝑄𝑞 is the generating function of a canonical transformation from
(𝑝, 𝑞) to (𝑃, 𝑄), then which one of the following relations is correct?

𝑝 𝑄
(A) =
𝑃 𝑞

𝑃 𝑄
(B) =
𝑝 𝑞

𝑝 𝑄
(C) =−
𝑃 𝑞

𝑃 𝑄
(D) =−
𝑝 𝑞

Q.12 An unpolarized plane electromagnetic wave in a dielectric medium 1 is incident on
a plane interface that separates medium 1 from another dielectric medium 2.
Medium 1 and medium 2 have refractive indices 𝑛1 and 𝑛2 , respectively, with
𝑛
𝑛2 > 𝑛1 . If the angle of incidence is tan−1 (𝑛2 ), which one of the following
1
statements is true?

(A) The reflected wave is unpolarized

(B) The reflected wave is polarized parallel to the plane of incidence

(C) The reflected wave is polarized perpendicular to the plane of incidence

(D) There is no transmitted wave

Page 8 of 44

Q.13 The wavefunction of a particle in an infinite one-dimensional potential well at time
𝑡 is

2 1 𝑖𝜋⁄6 −𝑖𝐸 𝑡⁄ℏ 1 𝑖𝜋⁄4 −𝑖𝐸 𝑡⁄ℏ
Ψ(𝑥, 𝑡) = √ 𝑒 −𝑖𝐸1 𝑡⁄ℏ 𝜓1 (𝑥) + 𝑒 𝑒 2 𝜓2 (𝑥) + 𝑒 𝑒 3 𝜓3 (𝑥)
3 √6 √6

where 𝜓1 , 𝜓2 and 𝜓3 are the normalized ground state, the normalized first excited
state and the normalized second excited state, respectively. 𝐸1 , 𝐸2 and 𝐸3 are the
eigen-energies corresponding to 𝜓1 , 𝜓2 and 𝜓3 , respectively. The expectation value
of energy of the particle in state Ψ(𝑥, 𝑡) is

(A) 17
𝐸
6 1

(B) 2
𝐸
3 1

(C) 3
𝐸
2 1

(D) 14 𝐸1

Page 9 of 44

Q.14 If a thermodynamical system is adiabatically isolated and experiences a change in
volume under an externally applied constant pressure, then the thermodynamical
potential minimized at equilibrium is the

(A) enthalpy

(B) Helmholtz free energy

(C) Gibbs free energy

(D) grand potential

Q.15 The mean distance between the two atoms of HD molecule is 𝑟, where H and D
denote hydrogen and deuterium, respectively. The mass of the hydrogen atom is
𝑚𝐻 . The energy difference between two lowest lying rotational states of HD in
multiples of ℏ2 /(𝑚𝐻 𝑟 2 ) is

3
(A)
2

2
(B)
3

(C) 6

4
(D)
3

Page 10 of 44

Q.16 Crystal structures of two metals A and B are two-dimensional square lattices with
same lattice constant a. Electrons in metals behave as free electrons. The Fermi
surfaces corresponding to A and B are shown by solid circles in figures.

The electron concentrations in A and B are 𝑛𝐴 and 𝑛𝐵 , respectively. The value of
𝑛
( 𝐵 ) is
𝑛𝐴

(A) 3

(B) 2

(C) 3√3

(D) √2

Page 11 of 44

Q.17 Consider the induced nuclear fission reaction
235 93 141
92𝑈 + 𝑛 → 37𝑅𝑏 + 55𝐶𝑠 + 2𝑛

where neutron momenta in both initial and final states are negligible. The ratio of
the kinetic energies (KE) of the daughter nuclei,
93
KE( 37𝑅𝑏)
KE( 141
55𝐶𝑠)

is ___________

(A) 93
141

(B) 141
93

(C) 1

(D) 0

Page 12 of 44

Q.18 The symbols C, D, 𝑉𝑖𝑛 and 𝑉0 shown in the figure denote capacitor, ideal diode,
input voltage and output voltage, respectively.

Which one of the following output waveforms (𝑉0) is correct for the given input
waveform (𝑉𝑖𝑛 )?

(A)

(B)

(C)

(D)

Page 13 of 44

Q.19 Let 𝑁𝑒 and 𝑇𝑒 , respectively, denote number and kinetic energy of electrons produced
in a nuclear beta decay. Which one of the following distributions is correct?

(A)

(B)

(C)

(D)

Page 14 of 44

Q.20 An infinitely long cylinder of radius 𝑅 carries a frozen-in magnetization
⃗⃗ = 𝑘𝑒 −𝑠 𝑧̂ , where 𝑘 is a constant and 𝑠 is the distance from the axis of cylinder.
𝑀
The magnetic permeability of free space is 𝜇0 . There is no free current present
anywhere. The magnetic flux density (𝐵 ⃗ ) inside the cylinder is

(A) 0

(B) 𝜇0 𝑘𝑒 −𝑅 𝑧̂

(C) 𝜇0 𝑘𝑒 −𝑠 𝑧̂

(D) 𝑅
𝜇0 𝑘𝑒 −𝑠 ( ) 𝑧̂
𝑠

Q.21 Atomic numbers of V, Cr, Fe and Zn are 23, 24, 26 and 30, respectively. Which one
of the following materials does NOT show an electron spin resonance (ESR)
spectra?

(A) V

(B) Cr

(C) Fe

(D) Zn

Page 15 of 44

Q.22 A particle is subjected to a potential

∞, 𝑥≤0
𝑉(𝑥) = {𝑉0 , 𝑎 ≤ 𝑥 ≤ 𝑏
0, elsewhere

Here, 𝑎 > 0 and 𝑏 > 𝑎. If the energy of the particle 𝐸 < 𝑉0, which one of the
following schematics is a valid quantum mechanical wavefunction (Ψ) for the
system?

(A)

(B)

(C)

(D)

Page 16 of 44

Q.23 Let 𝜌(𝑝, 𝑞 , 𝑡) be the phase space density of an ensemble of a system.
The Hamiltonian of the system is 𝐻(𝑝, 𝑞 ). If {𝐴, 𝐵} denotes the Poisson bracket of
𝐴 and 𝐵, then

𝑑𝜌
=0
𝑑𝑡
implies

(A) 𝜕𝜌
=0
𝜕𝑡

(B) 𝜕𝜌
∝ {𝜌, 𝐻}
𝜕𝑡

(C) 𝜕𝜌 𝑝∙𝑞
∝ {𝜌, }
𝜕𝑡 2

(D) 𝜕𝜌 𝑞∙𝑞
∝ {𝜌, }
𝜕𝑡 2

Page 17 of 44

Q.24 Consider the following circuit:

Suppose the input signal 𝑃 is

and the input signal 𝑄 is

Which one of the following output signals is correct?

(A)

(B)

(C)

(D)

Page 18 of 44

Q.25 An inertial observer sees two spacecrafts S and T flying away from each other along
𝑥-axis with individual speed 0.5𝑐, where 𝑐 is the speed of light. The speed of T with
respect to S is

(A) 4
𝑐
5

(B) 4
𝑐
3

(C) 𝑐

(D) 2
𝑐
3

Q.26 Let P, Q and R be three different nuclei. Which one of the following nuclear
processes is possible?

(A) 𝜈𝑒 + 𝐴𝑧𝑃 → 𝑧+1𝐴𝑄 + 𝑒 −

(B) 𝜈𝑒 + 𝐴𝑧𝑃 → 𝑧−1𝐴𝑅 + 𝑒 +

(C) 𝜈𝑒 + 𝐴𝑧𝑃 → 𝐴𝑧𝑃 + 𝑒 + + 𝑒 −

(D) 𝜈𝑒 + 𝐴𝑧𝑃 → 𝐴𝑧𝑃 + 𝛾

Page 19 of 44

Q.27 The temperature dependence of the electrical conductivity (𝜎) of three intrinsic
semiconductors A, B and C is shown in figure.

Let EA , EB and EC be the bandgaps of A, B and C, respectively. Which one of the
following relations is correct?

(A) EC > EA > EB

(B) EB > EC > EA

(C) EA > EB > EC

(D) EA > EC > EB

Page 20 of 44

Q.28 Following trial wavefunctions
′
𝜙1 = 𝑒 −𝑍 (𝑟1 +𝑟2)

and
′
𝜙2 = 𝑒 −𝑍 (𝑟1 +𝑟2) (1 + 𝑔|𝑟1 − 𝑟2 |)

are used to get a variational estimate of the ground state energy of the helium atom.
𝑍 ′ and 𝑔 are the variational parameters, 𝑟1and 𝑟2 are the position vectors of the
electrons. Let 𝐸0 be the exact ground state energy of the helium atom. 𝐸1 and 𝐸2 are
the variational estimates of the ground state energy of the helium atom
corresponding to 𝜙1 and 𝜙2 , respectively. Which one of the following options is
true?

(A) 𝐸1 ≤ 𝐸0 , 𝐸2 ≤ 𝐸0 , 𝐸1 ≥ 𝐸2

(B) 𝐸1 ≥ 𝐸0 , 𝐸2 ≤ 𝐸0 , 𝐸1 ≥ 𝐸2

(C) 𝐸1 ≤ 𝐸0 , 𝐸2 ≥ 𝐸0 , 𝐸1 ≤ 𝐸2

(D) 𝐸1 ≥ 𝐸0 , 𝐸2 ≥ 𝐸0 , 𝐸1 ≥ 𝐸2

Q.29 The wavefunction for a particle is given by the form 𝑒 −(𝑖𝛼𝑥+𝛽) , where 𝛼 and 𝛽 are
real constants. In which one of the following potentials 𝑉(𝑥), the particle is moving?

(A) 𝑉(𝑥) ∝ 𝛼 2 𝑥 2

(B) 𝑉(𝑥) ∝ 𝑒 −𝛼𝑥

(C) 𝑉(𝑥) = 0

(D) 𝑉(𝑥) ∝ sin(𝛼𝑥)

Page 21 of 44

Q.30 Consider a volume integral

1
𝐼 = ∫ ∇2 ( ) 𝑑𝑉
𝑉 𝑟

over a volume 𝑉, where 𝑟 = √𝑥 2 + 𝑦 2 + 𝑧 2 . Which of the following statement
is/are correct?

(A) 𝐼 = −4𝜋, if 𝑟 = 0 is inside the volume 𝑉

(B) Integrand vanishes for 𝑟 ≠ 0

(C) 𝐼 = 0, if 𝑟 = 0 is not inside the volume 𝑉

(D) Integrand diverges as 𝑟 → ∞

Q.31 The complex function

2
−(𝑧−1)
𝑒
has __________

(A) a simple pole at 𝑧 = 1

(B) an essential singularity at 𝑧 = 1

(C) a residue equal to −2 at 𝑧 = 1

(D) a branch point at 𝑧 = 1

Page 22 of 44

Q.32 The minimum number of basic logic gates required to realize the Boolean
expression 𝐵 ∙ (𝐴 + 𝐵) + 𝐴 ∙ (𝐵̅ + 𝐴) is _____ (in integer).

Q.33 3754
The vapor pressure (P) of solid ammonia is given by ln(𝑃) = 23.03 − , while
𝑇
3063
that of liquid ammonia is given by ln(𝑃) = 19.49 − , where 𝑇 is the
𝑇
temperature in K.

The temperature of the triple point of ammonia is _____ K (rounded off to two
decimal places).

Q.34 The electric field in a region depends only on 𝑥 and 𝑦 coordinates as

(𝑥 𝑥̂ + 𝑦 𝑦̂ )
𝐸⃗ = 𝑘
𝑥2 + 𝑦2

where 𝑘 is a constant. The flux of 𝐸⃗ through the surface of a sphere of radius 𝑅 with
its center at the origin is 𝑛𝜋𝑘𝑅, where the value of 𝑛 is _____ (in integer).

Q.35 The Hamiltonian of a system of N particles in volume V at temperature T is
2𝑁 2𝑁

𝐻 = ∑ 𝑎𝑖 𝑞𝑖2 + ∑ 𝑏𝑖 𝑝𝑖2
𝑖=1 𝑖=1

where 𝑎𝑖 and 𝑏𝑖 are positive constants. The ensemble average of the Hamiltonian is
𝛼𝑁𝑘𝐵 𝑇, where 𝑘𝐵 is the Boltzmann constant. The value of 𝛼 is ____ (in integer).

Page 23 of 44

Q.36 – Q.65 Carry TWO marks Each

Q.36 Binding energy and rest mass energy of a two-nucleon bound state are denoted by
𝐵 and 𝑚𝑐 2 , respectively, where 𝑐 is the speed of light. The minimum energy of a
photon required to dissociate the bound state is

(A) 𝐵

(B) 𝐵
𝐵 (1 + )
2𝑚𝑐 2

(C) 𝐵
𝐵 (1 − )
2𝑚𝑐 2

(D) 𝐵 − 𝑚𝑐 2

Q.37 The spin-orbit interaction in a hydrogen-like atom is given by the Hamiltonian

⃗ ∙𝑆
𝐻 ′ = −𝑘𝐿

where 𝑘 is a real constant. The splitting between levels 2𝑝3/2 and 2𝑝1/2 due to this
interaction is

(A) 1 2
𝑘ℏ
2

(B) 3 2
𝑘ℏ
2

(C) 3 2
𝑘ℏ
4

(D) 2𝑘ℏ2

Page 24 of 44

Q.38 Consider the Lagrangian 𝐿 = 𝑚𝑥̇ 𝑦̇ − 𝑚𝜔02 𝑥𝑦. If 𝑝𝑥 and 𝑝𝑦 denote the generalized
momenta conjugate to 𝑥 and 𝑦, respectively, then the canonical equations of motion
are

(A) 𝑝𝑥 𝑝𝑦
𝑥̇ = , 𝑝̇𝑥 = −𝑚𝜔02 𝑥 , 𝑦̇ = , 𝑝̇𝑦 = −𝑚𝜔02 𝑦
𝑚 𝑚

(B) 𝑝𝑥 𝑝𝑦
𝑥̇ = , 𝑝̇𝑥 = 𝑚𝜔02 𝑥 , 𝑦̇ = , 𝑝̇𝑦 = 𝑚𝜔02 𝑦
𝑚 𝑚

(C) 𝑝𝑦 𝑝𝑥
𝑥̇ = , 𝑝̇𝑥 = −𝑚𝜔02 𝑦 , 𝑦̇ = , 𝑝̇𝑦 = −𝑚𝜔02 𝑥
𝑚 𝑚

(D) 𝑝𝑦 𝑝𝑥
𝑥̇ = , 𝑝̇𝑥 = 𝑚𝜔02 𝑦 , 𝑦̇ = , 𝑝̇𝑦 = 𝑚𝜔02 𝑥
𝑚 𝑚

Q.39 The X-ray diffraction pattern of a monatomic cubic crystal with rigid spherical
atoms of radius 1.56 Å shows several Bragg reflections of which the reflection
appearing at the lowest 2𝜃 value is from (111) plane. If the wavelength of X-ray
used is 0.78 Å, the Bragg angle (in 2𝜃, rounded off to one decimal place)
corresponding to this reflection and the crystal structure, respectively, are

(A) 21.6 and body centered cubic

(B) 17.6 and face centered cubic

(C) 10.8 and body centered cubic

(D) 8.8 and face centered cubic

Page 25 of 44

Q.40 In a parallel plate capacitor, the plate at 𝑥 = 0 is grounded and the plate at 𝑥 = 𝑑 is
maintained at a potential 𝑉0. The space between the two plates is filled with a linear
𝑥
dielectric of permittivity 𝜖 = 𝜖0 (1 + 𝑑), where 𝜖0 is the permittivity of free space.
Neglecting the edge effects, the electric field (𝐸⃗ ) inside the capacitor is

(A) 𝑉0
− 𝑥̂
(𝑑 + 𝑥) ln 2

(B) 𝑉0
− 𝑥̂
𝑑

(C) 𝑉0
− 𝑥̂
(𝑑 + 𝑥)

(D) 𝑉0 𝑑
− 𝑥̂
(𝑑 + 𝑥)𝑥

Q.41 The equation of motion for the forced simple harmonic oscillator is

𝑥̈ (𝑡) + 𝜔2 𝑥(𝑡) = 𝐹 cos(𝜔𝑡)

where 𝑥(𝑡 = 0) = 0 and 𝑥̇ (𝑡 = 0) = 0. Which one of the following options is
correct?

(A) 𝑥(𝑡) ∝ 𝑡 sin(𝜔𝑡)

(B) 𝑥(𝑡) ∝ 𝑡 cos(𝜔𝑡)

(C) 𝑥(𝑡) = ∞

(D) 𝑥(𝑡) ∝ 𝑒 𝜔𝑡

Page 26 of 44

Q.42 An atom is subjected to a weak uniform magnetic field 𝐵 ⃗ . The number of lines in
its Zeeman spectrum for transition from 𝑛 = 2, 𝑙 = 1 to 𝑛 = 1, 𝑙 = 0 is

(A) 8

(B) 10

(C) 12

(D) 5

Q.43 1 2 1 0
Consider two matrices: P = [ ] and Q = [ ].
0 1 0 1
Which of the following statement is/are true?

(A) P and Q have same set of eigenvalues

(B) P and Q commute with each other

(C) P and Q have different sets of linearly independent eigenvectors

(D) P is diagonalizable

Page 27 of 44

Q.44 An infinite one dimensional lattice extends along 𝑥-axis. At each lattice site there
1
exists an ion with spin 2 . The spin can point either in +𝑧 or – 𝑧 direction only. Let
SP, SF, and SA denote the entropies of paramagnetic, ferromagnetic and
antiferromagnetic configurations, respectively. Which of the following relation
is/are true?

(A) SP > SF

(B) SA > S F

(C) SA = 4 S F

(D) SP > SA

Page 28 of 44

Q.45 Consider a vector field 𝐹 = (2𝑥𝑧 + 3𝑦 2 )𝑦̂ + 4𝑦𝑧 2 𝑧̂ . The closed path
(Γ: A → B → C → D → A) in 𝑧 = 0 plane is shown in figure.

∮Γ 𝐹 . ⃗⃗⃗
𝑑𝑙 denotes the line integral of 𝐹 along the closed path Γ. Which of the
following option is/are true?

MSQ

(A)
∮ 𝐹 . ⃗⃗⃗
𝑑𝑙 = 0
Γ

(B) 𝐹 is non-conservative

(C) ⃗∇. 𝐹 = 0

(D) 𝐹 can be written as the gradient of a scalar field

Page 29 of 44

Q.46 Two point charges of charge +𝑞 each are placed a distance 2𝑑 apart. A grounded
solid conducting sphere of radius 𝑎 is placed midway between them.
Assume 𝑎2 ≪ 𝑑2 . Which of the following statement is/are true?

(A) 𝑑
If 𝑎 > 8 , the net force acting on the charges is directed towards each other

(B) The potential at the surface of the sphere is zero

(C) 2𝑎𝑞
Total induced charge on the sphere is (− 𝑑 )

(D) The potential at the center of the sphere is non-zero

Page 30 of 44

Q.47 A particle of mass 𝑚 is moving in the potential

1
𝑉(𝑥) = { 𝑉0 + 𝑚𝜔02 𝑥 2 , 𝑥 > 0,
2
∞, 𝑥 ≤ 0,

Figures P, Q, R and S show different combinations of the values of 𝜔0 and 𝑉0.

(P) (Q) (R) (S)
𝐸𝑗 , 𝐸𝑗 , 𝐸𝑗 and 𝐸𝑗 with 𝑗 = 0, 1, 2, …, are the eigen-energies of the 𝑗-th level
for the potentials shown in figures P, Q, R and S, respectively. Which of the
statement is/are true?

(A) (P) (Q)
𝐸0 = 𝐸0

(B) (Q) (S)
𝐸0 = 𝐸0

(C) (P) (R)
𝐸0 = 𝐸1

(D) (R) (Q)
𝐸0 ≠ 𝐸0

Page 31 of 44

Q.48 The non-relativistic Hamiltonian for a single electron atom is

𝑝2
𝐻0 = − 𝑉(𝑟)
2𝑚
where 𝑉(𝑟) is the Coulomb potential and 𝑚 is the mass of the electron. Considering
the spin-orbit interaction term

1 1 𝑑𝑉
𝐻′ = ⃗ ∙𝑆
𝐿
2𝑚2 𝑐 2 𝑟 𝑑𝑟
added to 𝐻0 , which of the following statement is/are true?

(A) 𝐻′ commutes with 𝐿2

(B) 𝐻′ commutes with 𝐿𝑧 and 𝑆𝑧

(C) For a given value of principal quantum number 𝑛 and orbital angular momentum
quantum number 𝑙, there are 2(2𝑙 + 1) degenerate eigenstates of 𝐻0

(D) 𝐻0 , 𝐿2 , 𝑆 2 , 𝐿𝑧 and 𝑆𝑧 have a set of simultaneous eigenstates

Q.49 Decays of mesons and baryons can be categorized as weak, strong and
electromagnetic decays depending upon the interactions involved in the processes.
Which of the following option is/are true?

(A) 𝜋 0 → 𝛾𝛾 is a weak decay

(B) ∧0 → 𝜋 0 + 𝑝 is an electromagnetic decay

(C) 𝐾0 → 𝜋+ + 𝜋− is a weak decay

(D) ∆++ → 𝑝 + 𝜋 + is a strong decay

Page 32 of 44

Q.50 An extrinsic semiconductor shown in figure carries a current of 2 mA along its
length parallel to +𝑥 axis.

When the majority charge carrier concentration is 12.5 × 1013 cm−3 and the
sample is exposed to a constant magnetic field applied along the +𝑧 direction, a
Hall voltage of 20 mV is measured with the negative polarity at 𝑦 = 0 plane.
Take the electric charge as 1.6 × 10−19 C. The concentration of minority charge
carrier is negligible. Which of the following statement is/are true?

(A) The majority charge carrier is electron

(B) The magnitude of the applied magnetic field is 1 Tesla

(C) The electric field corresponding to the Hall voltage is in the +𝑦 direction

(D) The magnitude of Hall coefficient is 50,000 m3 C -1

Page 33 of 44

Q.51 𝐴𝛼 and 𝐵𝛽 (𝛼, 𝛽 = 1,2,3, ⋯ , 𝑛) are contravariant and covariant vectors,
respectively. By convention, any repeated indices are summed over. Which of the
following expression is/are tensors?

(A) 𝐴𝛼 𝐵𝛽

(B) 𝐴𝛼 𝐵𝛽
𝐴𝛼 𝐵𝛼

(C) 𝐴𝛼
𝐵𝛽

(D) 𝐴𝛼 + 𝐵𝛽

Page 34 of 44

Q.52 The temperature T dependence of magnetic susceptibility 𝜒 (Column I) of certain
magnetic materials (Column II) are given below. Which of the following option
is/are correct?

Column I Column II

(1) (P) Diamagnetic

(2) (Q) Paramagnetic

(3) (R) Ferromagnetic

(4) (S) Antiferromagnetic

(A) 2 − P, 4 − Q, 3 − S

(B) 4 − P, 1 − Q, 2 − R

(C) 4 − Q, 2 − R, 1 − S

(D) 3 − P, 4 − Q, 2 − R

Page 35 of 44

Q.53 The curves P and Q schematically show the variation of X-ray intensity with
wavelength at two different accelerating voltages for a given target material. In the
figure 𝜆1 = 0.25 Å, 𝜆2 = 0.5 Å, 𝜆3 = 1.0 Å, and 𝜆4 = 2.25 Å. Take Planck’s
constant as 6.6 × 10−34 Js, speed of light as 3 × 108 ms-1 and elementary charge
as 1.6 × 10−19 C.

Which of the following statement is/are true?

(A) The accelerating potential corresponding to curve P is greater than that of curve Q

(B) The accelerating potential applied to obtain curve Q is 24750 V

(C) Peaks (II) and (IV) correspond to radiative transitions from L to K shells

(D) Peaks (I) and (III) correspond to radiative transitions from N to K shells

Page 36 of 44

Q.54 Apart from the acoustic modes, 9 optical modes are identified from the
measurements of phonon dispersions of a solid with chemical formula 𝐴𝑛 𝐵𝑚 , where
𝐴 and 𝐵 denote the atomic species, and 𝑛 and 𝑚 are integers. Which of the following
combination of n and m is/are possible?

(A) 𝑛 = 1, 𝑚 = 1

(B) 𝑛 = 2, 𝑚 = 2

(C) 𝑛 = 3, 𝑚 = 1

(D) 𝑛 = 4, 𝑚 = 4

Page 37 of 44

Q.55 An oscillating electric dipole of moment 𝑑 (𝑡) = 𝑑0 cos(𝜔𝑡) 𝑧̂ is placed at origin as
shown in figure.

Consider a point P(𝑟, 𝜃, 𝜙) at a very large distance from the dipole. Here 𝑟, 𝜃 and
𝜙 are spherical polar coordinates. Which of the following statement is/are true for
intensity of radiation?

(A) Intensity is zero if P is on the 𝑧 axis

(B) 𝜋 𝜋
Intensity is zero at P (𝑟 = 𝑅, 𝜃 = 2 , 𝜙 = 4 )

(C) 𝜋 𝜋
Intensity at P (𝑟 = 𝑅, 𝜃 = 2 , 𝜙 = 4 ) is greater than that
𝜋 𝜋
at P (𝑟 = 𝑅, 𝜃 = 4 , 𝜙 = 4 )

(D) 𝜋 𝜋 𝜋 𝜋
Intensity at P (𝑟 = 𝑅, 𝜃 = 2 , 𝜙 = 4 ) is equal to that at P (𝑟 = 𝑅, 𝜃 = 4 , 𝜙 = 4 )

Page 38 of 44

Q.56 The Fourier transform and its inverse transform are respectively defined as
1 +∞ 1 +∞
𝑓̃ (𝜔) = ∫−∞ 𝑓(𝑥)𝑒 𝑖𝜔𝑥 𝑑𝑥 and 𝑓(𝑥) = ∫−∞ 𝑓̃ (𝜔)𝑒 −𝑖𝜔𝑥 𝑑𝜔. Consider two
√2𝜋 √2𝜋
functions 𝑓 and 𝑔. Another function 𝑓 ∗ 𝑔 is defined as
+∞
1
(𝑓 ∗ 𝑔)(𝑥) = ∫ 𝑓(𝑦)𝑔(𝑥 − 𝑦)𝑑𝑦
√2𝜋 −∞

Which of the following relation is/are true?

Note: Tilde (~) denotes the Fourier transform.

(A) 𝑓∗𝑔 = 𝑔∗𝑓

(B) 𝑓̃
∗ 𝑔 = 𝑔̃
∗𝑓

(C) 𝑓̃ ̃
∗ 𝑔 = 𝑓𝑔

(D) 𝑓̃
∗ 𝑔 = 𝑓̃𝑔̃

Page 39 of 44

Q.57 A material behaves as a superconductor below a critical temperature 𝑇𝑐 and as a
normal conductor above 𝑇𝑐 . A magnetic field 𝐵 ⃗ = 𝐵𝑧̂ is applied when 𝑇 > 𝑇𝑐 .
The material is then cooled below 𝑇𝑐 in the presence of 𝐵 ⃗ . Which of the following
figure represent the correct configuration of magnetic field lines?

(A)

(B)

(C)

(D)

Page 40 of 44

Q.58 A typical biasing of a silicon transistor is shown in figure.

The value of common-emitter current gain β for the transistor is 100. Ignore reverse
saturation current. The output voltage 𝑉0 (in V) is ____ (in integer).

Q.59 The canonical partition function of an ideal gas is
𝑁
1 𝑉
𝑄(𝑇, 𝑉, 𝑁) = [ ]
𝑁! (𝜆(𝑇))3

where 𝑇, 𝑉, 𝑁 and 𝜆(𝑇) denote temperature, volume, number of particles, and
thermal de Broglie wavelength, respectively. Let 𝑘𝐵 be the Boltzmann constant and
𝜇 be the chemical potential. Take ln(𝑁!) = 𝑁 ln(𝑁) − 𝑁.
𝑁
If the number density ( 𝑉 ) is 2.5 × 1025 m−3 at a temperature T,
𝑒 𝜇/(𝑘𝐵𝑇)
then
(𝜆(𝑇))3
× 10−25 is _______ m−3 (rounded off to one decimal place).

Page 41 of 44

Q.60 1
Lagrangian of a particle of mass 𝑚 is 𝐿 = 2 𝑚𝑥̇ 2 − 𝜆𝑥 4 , where 𝜆 is a positive
constant. If the particle oscillates with total energy 𝐸, then the time period of
oscillations is
1
𝐸 4
( )
𝜆 𝑑𝑥
𝑎∫
0
√( 2 ) (𝐸 − 𝜆𝑥 4 )
𝑚

The value of 𝑎 is _____ (in integer).

Q.61 A particle of mass 𝑚 in an infinite potential well of width 𝑎 is subjected to a
ℎ2
perturbation, 𝑉 ′ = as shown in figure, where ℎ is Planck’s constant.
40𝑚𝑎2

The first order energy shift of the fourth energy eigenstate due to this perturbation
is

ℎ2
( )
𝑁𝑚𝑎2

The value of 𝑁 is _____ (in integer).

Page 42 of 44

Q.62 Consider a three-dimensional system of non-interacting bosons with zero chemical
potential. The energy of the system 𝜖 ∝ 𝑘 2 , where 𝑘 is the wavevector. The low
temperature specific heat of the system at constant volume depends on the
𝑛
temperature as 𝐶𝑉 ∝ 𝑇 2 . The value of 𝑛 is _____ (in integer).

Q.63 Consider the operational amplifier circuit shown in figure.

The output voltage 𝑉0 is _____ V (in integer).

Page 43 of 44

Q.64 An electron in the Coulomb field of a proton is in the following state of coherent
superposition of orthonormal states 𝜓𝑛𝑙𝑚

1 1 √5
Ψ = 𝜓100 + 𝜓210 − 𝜓
3 √3 3 320

Let 𝐸1 , 𝐸2 , and 𝐸3 represent the first three energy levels of the system. A sequence
of measurements is done on the same system at different times. Energy is measured
first at time 𝑡1 and the outcome is 𝐸2 . Then total angular momentum is measured at
time 𝑡2 > 𝑡1 and finally energy is measured again at 𝑡3 > 𝑡2 . The probability of
finding the system in a state with energy 𝐸2 after the final measurement is 𝑃/9.
The value of 𝑃 is _____ (in integer).

Q.65 According to the nuclear shell model, the absolute value of the difference in
magnetic moments of 158𝑂 and 157𝑁 , in the units of nuclear magneton (𝜇𝑁 ) is 𝑎/3.

The magnitude of 𝑎 is _____ (in integer).

Page 44 of 44

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