GATE 2025 Question Paper Physics (PH)

GATE
2025
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
Q.1 – Q.5 Carry ONE mark Each

Q.1 Is there any good show _______ television tonight?

Select the most appropriate option to complete the above sentence.

(A) in

(B) at

(C) within

(D) on

Q.2 As the police officer was found guilty of embezzlement, he was _______ dismissed
from the service in accordance with the Service Rules.

Select the most appropriate option to complete the above sentence.

(A) sumptuously

(B) brazenly

(C) unintentionally

(D) summarily

Page 1 of 39

Q.3 The sum of the following infinite series is:

1 1 1 1 1
+ + + + +⋯
1! 2! 3! 4! 5!

(A) 𝜋

(B) 1+𝑒

(C) 𝑒−1

(D) 𝑒

Q.4 A thin wire is used to construct all the edges of a cube of 1 m side by bending,
cutting and soldering the wire. If the wire is 12 m long, what is the minimum number
of cuts required to construct the wire frame to form the cube?

(A) 3

(B) 4

(C) 6

(D) 12

Page 2 of 39

Q.5 The figures I, II and III are parts of a sequence. Which one of the following options
comes next in the sequence at IV?

?

I II III IV

(A)

(B)

(C)

(D)

Page 3 of 39

Q.6 – Q.10 Carry TWO marks Each

Q.6 “Why do they pull down and do away with crooked streets, I wonder, which are my
delight, and hurt no man living? Every day the wealthier nations are pulling down
one or another in their capitals and their great towns: they do not know why they do
it; neither do I. It ought to be enough, surely, to drive the great broad ways which
commerce needs and which are the life-channels of a modern city, without
destroying all history and all the humanity in between: the islands of the past.”

(From Hilaire Belloc’s “The Crooked Streets”)

Based only on the information provided in the above passage, which one of the
following statements is true?

(A) The author of the passage takes delight in wondering.

(B) The wealthier nations are pulling down the crooked streets in their capitals.

(C) In the past, crooked streets were only built on islands.

(D) Great broad ways are needed to protect commerce and history.

Page 4 of 39

Q.7 Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant,
he notices that the hour and minute hands on the wall clock are exactly coinciding.
After about an hour, when he leaves the restaurant, he notices that the clock hands
are again exactly coinciding. How much time (in minutes) did Rohit spend at the
restaurant?

(A) 6
64 11

(B) 5
66 13

(C) 5
65 11

(D) 6
66 13

Page 5 of 39

Q.8 A color model is shown in the figure with color codes: Yellow (Y), Magenta (M),
Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K).

Which one of the following options displays the color codes that are consistent with
the color model?

(A)

(B)

(C)

(D)

Page 6 of 39

Q.9 A circle with center at (𝑥, 𝑦) = (0.5, 0) and radius = 0.5 intersects with another
circle with center at (𝑥, 𝑦) = (1, 1) and radius = 1 at two points. One of the points
of intersection (𝑥, 𝑦) is:

(A) (0, 0)

(B) (0.2, 0.4)

(C) (0.5, 0.5)

(D) (1, 2)

Page 7 of 39

Q.10 An object is said to have an 𝑛-fold rotational symmetry if the object, rotated by an
2𝜋
angle of , is identical to the original.
𝑛

Which one of the following objects exhibits 4-fold rotational symmetry about an
axis perpendicular to the plane of the screen?

Note: The figures shown are representative.

(A)

(B)

(C)

(D)

Page 8 of 39

Q.11 – Q.35 Carry ONE mark Each

Q.11 For a two-dimensional hexagonal lattice with lattice constant 𝑎, the atomic density
is

(A) 1
√3𝑎2

(B) 1
√6𝑎2

(C) 4
3√3𝑎2

(D) 1
3√3𝑎2

Q.12 Consider a crystal that has a basis of one atom. Its primitive vectors are ⃗⃗⃗⃗
𝑎1 = 𝑎𝑖̂ ,
𝑎
𝑎2 = 𝑎𝑗̂ , ⃗⃗⃗⃗
⃗⃗⃗⃗ ̂ ̂
𝑎3 = (𝑖̂ + 𝑗̂ + 𝑘 ), where 𝑖̂, 𝑗̂, 𝑘 are the unit vectors in the 𝑥, 𝑦 and 𝑧
2
directions of the Cartesian coordinate system and 𝑎 is a positive constant. Which
one of the following is the correct option regarding the type of the Bravais lattice?

(A) 𝑎3
It is BCC and the volume of the primitive cell is
2

(B) 𝑎3
It is FCC and the volume of the primitive cell is
4

(C) 𝑎3
It is BCC and the volume of the primitive cell is
8

(D) It is FCC and the volume of the primitive cell is 𝑎3

Page 9 of 39

Q.13 1
A particle of mass m is in a potential 𝑉(𝑥) = 2 𝑚𝜔2 𝑥 2 for 𝑥 > 0 and 𝑉(𝑥) = ∞ for
𝑥 ≤ 0, where 𝜔 is the angular frequency. The ratio of the energies corresponding
to the lowest energy level to the next higher level is

(A) 3
7

(B) 1
2

(C) 1
3

(D) 3
5

Q.14 A particle is scattered from a potential 𝑉(𝑟) = 𝑔𝛿 3 (𝑟), where 𝑔 is a positive
constant. Using the first Born approximation, the angular (𝜃, 𝜙) dependence of
𝑑𝜎
differential scattering cross section is
𝑑Ω

(A) Independent of 𝜃 but dependent on 𝜙

(B) Dependent on 𝜃 but independent of 𝜙

(C) Dependent on both 𝜃 and 𝜙

(D) Independent of both 𝜃 and 𝜙

Page 10 of 39

Q.15 The Joule-Thomson expansion of a gas is

(A) Isentropic

(B) Isenthalpic

(C) Isobaric

(D) Isochoric

Q.16 Which one of the following is correct for the phase velocity 𝑣𝑝 and group velocity
𝑣𝑔 ? (𝑐 is the speed of light in vacuum)

(A) For matter waves in the relativistic case, 𝑣𝑝 𝑣𝑔 > 𝑐 2

(B) For electromagnetic waves in a medium, 𝑣𝑝 represents the speed with which energy
propagates

(C) For electromagnetic waves in a medium, both 𝑣𝑝 and 𝑣𝑔 can be more than 𝑐

(D) For matter waves in free space, 𝑣𝑝 ≠ 𝑣𝑔

Page 11 of 39

Q.17 As per the Drude model of metals, the electrical resistance of a metallic wire of
length 𝐿 and cross-section area 𝐴 is

(Consider 𝜏 as the relaxation time, 𝑚 as electron mass, 𝑛 as carrier concentration
and 𝑒 as electronic charge)

(A) 𝑚𝐿
𝑛𝑒 2 𝐴𝜏

(B) 2𝑚𝐿
𝑛𝑒 2 𝐴𝜏

(C) 𝑚𝐿
2𝑛𝑒 2 𝐴𝜏

(D) 𝑚𝐿
4𝑛𝑒 2 𝐴𝜏

Q.18 Which one of the following baryons has strangeness quantum number 𝑆 = −1?

(A) Σ ∗0

(B) 𝑛

(C) Ξ ∗0

(D) Δ0

Page 12 of 39

Q. 19 A logic gate circuit is shown in the figure below. The correct combination for the
input (𝑃, 𝑄) for which the output 𝑇 = 1 is

(A) (0,0)

(B) (0,1)

(C) (1,1)

(D) (1,0)

Q.20 The nuclear energy levels of mirror nuclei are similar. Using this empirical fact
alone, the nuclear force can be said to be independent of which one of the following
properties of the nucleons?

(A) Mass

(B) Spin

(C) Charge

(D) Parity

Page 13 of 39

Q.21 1
Consider the function 𝑓(𝑧) = of a complex variable 𝑧. The residues of
𝑧 2 (𝑧−2)3
the function at 𝑧 = 0 and 𝑧 = 2, respectively, are

(A) 3 3
− and
8 8

(B) 3 3
and −
8 16

(C) 3 3
− and
16 16

(D) 3 3
− and
8 16

Q.22 Consider one mole of a monovalent metal at absolute zero temperature, obeying the
free electron model. Its Fermi energy is 𝐸𝐹 . The energy corresponding to the filling
𝑁𝐴
of electrons, where 𝑁𝐴 is the Avogadro number, is 2𝑛 𝐸𝐹 . The value of 𝑛 is
2

(A) 2
−
3

(B) 2
+
3

(C) 1
−
3

(D) −1

Page 14 of 39

Q.23 A paramagnetic material containing paramagnetic ions with total angular
1
momentum 𝐽 = is kept at absolute temperature T. The ratio of the magnetic field
2
required for 80% of the ions to be in the lowest energy state to that required for
having 60% of the ions to be in the lowest energy state at the same temperature is

(A) 2ln2
3
ln (2)

(B) ln2
3
ln (2)

(C) 3ln2
3
ln (2)

(D) ln3
3
ln (2)

Q.24 Which of the following option(s) is/are correct for the ground state of a hydrogen
atom?

(A) Linear Stark effect is zero

(B) It has definite parity

(C) Spin-orbit coupling is zero

(D) Hyperfine splitting is zero

Page 15 of 39

Q.25 Which of the following option(s) is/are correct for photons?

(A) Its rest mass is zero, but its energy is non-zero

(B) It carries non-zero linear momentum

(C) It carries zero spin angular momentum

(D) It has two linearly independent states of polarization

Q.26 A schematic Pressure-Temperature diagram of water is shown in the figure. Which
of the following option(s) is/are correct?

(A) Clausius-Clapeyron equation is valid across the melting curve and the vaporization
curve

(B) Melting curve has the highest slope

(C) The critical point exists only for the vaporization curve

(D) Clausius-Clapeyron equation is not valid across the melting curve and the
vaporization curve

Page 16 of 39

Q.27 Which of the following consideration(s) is/are showing that nuclear beta decay,
𝑛 → 𝑝 + 𝑒 − + 𝜈̅𝑒 , has to be a three-body decay?

(A) Continuous distribution of the electron energy

(B) Spin of the final state

(C) Mass of the electron

(D) Mass of the proton

Q.28 Potential energy of two diatomic molecules P and Q of the same reduced mass is
shown in the figure. According to this diagram, which of the following option(s)
is/are correct?

(A) The equilibrium inter-nuclear distance of Q is more than that of P

(B) The total energy 𝐸 = 0 separates bound and unbound states of the molecules

(C) The lowest vibrational frequency of P is larger than that of Q

(D) Dissociation energy of Q is more than that of P

Page 17 of 39

Q.29 Nuclear radiation emitted from a 60Co radioactive source is detected by a
photomultiplier tube (PMT) coupled to a scintillator crystal. Which of the following
option(s) is/are correct?

(A) 𝛾 radiation from 60Co will directly hit the photocathode of the PMT without
interacting with the scintillator crystal and produce a signal

(B) 𝛽 radiation from 60Co source interacts with the scintillator crystal, producing 𝛾
radiation, which will hit the photocathode of the PMT and produce a signal

(C) A mu-metal shield is put all around the PMT to nullify the effect of external electric
fields

(D) A mu-metal shield is put all around the PMT to nullify the effect of external
magnetic fields

Q.30 One mole of an ideal monatomic gas at absolute temperature T undergoes free
expansion to double its original volume, so that the entropy change is Δ𝑆1 . An
identical amount of the same gas at absolute temperature 2T undergoes isothermal
expansion to double its original volume, so that the entropy change is Δ𝑆2. The
Δ𝑆1
value of (in integer) is ____
Δ𝑆2

Q.31 A linear dielectric sphere of radius R has a uniform frozen-in polarization along the
𝑧-axis. The center of the sphere initially coincides with the origin, about which the
electric dipole moment is 𝑝1. When the sphere is shifted to the point (2R, 0, 0), the
|𝑝1 |
corresponding dipole moment with respect to the origin is 𝑝2 . The value of
|𝑝2 |
(in integer) is _____

Page 18 of 39

Q.32 The effective magnetic moment (in units of Bohr magneton) for the ground state of
an isolated 4𝑓 ion with 6 unpaired electrons in the 4𝑓 shell according to Hund’s
rules is (in integer) _____

Q.33 Powder X-ray diffraction pattern of a cubic solid with lattice constant 𝑎 has the
(111) diffraction peak at 𝜃 = 30°. If the lattice expands such that the lattice constant
becomes 1.25𝑎, the angle (in degrees) corresponding to the (111) peak changes to
1
sin−1 ( ). The value of 𝑛 (rounded off to one decimal place) is _____
𝑛

Q.34 Consider a monatomic chain of length 30 cm. The phonon density of states is
1.2 × 10−4 s. Assuming the Debye model, the velocity of sound in m/s (rounded
off to one decimal place) is _____

Q.35 3
The Δ+ baryon with spin , at rest, decays to a proton and a pion (Δ+ → 𝑝 + 𝜋 0 ).
2
The Δ+ has positive intrinsic parity and 𝜋 0 has negative intrinsic parity. The orbital
angular momentum of the proton-pion system (in integer) is _____

Page 19 of 39

Q.36 – Q.55 Carry TWO marks Each

Q.36 The screened nuclear charge of neutral Helium atom is given as 1.7𝑒, where 𝑒 is
the magnitude of the electronic charge. Assuming the Bohr model of the atom for
𝑍2 1
which the energy levels are 𝐸𝑛 = − atomic units (𝑍 is the atomic number),
2 𝑛2
the first ionization potential of Helium in atomic units is

(A) 0.89

(B) 1.78

(C) 0.94

(D) 3.16

Page 20 of 39

Q.37 The wire loop shown in the figure carries a steady current 𝐼. Each straight section
of the loop has length 𝑑. A part of the loop lies in 𝑥𝑦 plane and the other part is
tilted at 30° with respect to the 𝑥𝑧 plane. The magnitude of the magnetic dipole
moment of the loop (in appropriate units) is

(A) √2 𝐼𝑑 2

(B) 2 𝐼𝑑2

(C) √3 𝐼𝑑 2

(D) 𝐼𝑑2

Page 21 of 39

Q.38 The figure shows a system of two equal masses 𝑚 and three massless horizontal
springs with spring constants 𝑘1 , 𝑘2 , 𝑘1 . Ignore gravity. The masses can move only
in the horizontal direction and there is no dissipation. If 𝑚 = 1, 𝑘1 = 2 and 𝑘2 = 3
(all in appropriate units), the frequencies of the normal modes of the system in the
same system of units are

(A) √2 , √8

(B) √2 , √6

(C) √3 , √10

(D) √3 , √8

Page 22 of 39

Q.39 Two projectile protons 𝑃1 and 𝑃2 both with spin up (along the +𝑧 direction) are
scattered from another fixed target proton 𝑇 with spin up at rest in the 𝑥𝑦 plane, as
shown in the figure. They scatter one at a time. The nuclear interaction potential
⃗ ⋅ 𝑆, where 𝐿
between both the projectiles and the target proton is 𝜆𝐿 ⃗ is the orbital
angular momentum of the system with respect to the target, 𝑆 is the spin angular
momentum of the system and 𝜆 is a negative constant in appropriate units. Which
one of the following is correct?

(A) 𝑃1 will be scattered in the +𝑦 direction (upward) and 𝑃2 will be scattered in the −𝑦
direction (downward)

(B) 𝑃1 will be scattered in the +𝑦 direction (upward) and 𝑃2 will be scattered in the
+𝑦 direction (upward)

(C) 𝑃1 will be scattered in the −𝑦 direction (downward) and 𝑃2 will be scattered in the
+𝑦 direction (upward)

(D) 𝑃1 will be scattered in the −𝑦 direction (downward) and 𝑃2 will be scattered in
the −𝑦 direction (downward)

Page 23 of 39

Q.40 A thin circular ring of radius R lies in the 𝑥𝑦 plane with its centre coinciding with
the origin. The ring carries a uniform line charge density 𝜆. The quadrupole
contribution to the electrostatic potential at the point (0, 0, 𝑑), where 𝑑 ≫ 𝑅, is

(A) 𝜆𝑅 3
−
4𝜀0 𝑑3

(B) 0

(C) 𝜆𝑅 3
4𝜀0 𝑑3

(D) 𝜆𝑅 3
−
2𝜀0 𝑑3

Page 24 of 39

Q.41 An 𝛼 particle is scattered from an Au target at rest as shown in the figure. 𝐷1 and
𝐷2 are the detectors to detect the scattered 𝛼 particle at an angle 𝜃 and along the
beam direction, respectively, as shown. The signals from 𝐷1 and 𝐷2 are converted
to logic signals and fed to logic gates. When a particle is detected, the signal is 1
and is 0 otherwise. Which one of the following circuits detects the particle scattered
at the angle 𝜃 only?

(A)

(B)

(C)

(D)

Page 25 of 39

Q.42 Consider a two-level system with energy states +𝜀 and – 𝜀 . The number of
particles at +𝜀 level is 𝑁+ and the number of particles at – 𝜀 level is 𝑁− . The total
energy of the system is 𝐸 and the total number of particles is 𝑁 = 𝑁+ + 𝑁− . In the
thermodynamic limit, the inverse of the absolute temperature of the system is
(Given: ln 𝑁! ≃ 𝑁 ln 𝑁 − 𝑁)

(A) 𝐸
𝑘𝐵 𝑁−
𝜀
ln [ 𝐸 ]
2𝜀 𝑁+
𝜀

(B) 𝑘𝐵
ln𝑁
𝜀

(C) 𝑘𝐵
ln𝑁
2𝜀

(D) 𝐸
𝑘𝐵 𝑁−
𝜀
ln [ 𝐸 ]
𝜀 𝑁+
𝜀

Q.43 Let |𝑚〉 and |𝑛〉 denote the energy eigenstates of a one-dimensional simple
harmonic oscillator. The position and momentum operators are 𝑋̂ and 𝑃̂,
respectively. The matrix element 〈𝑚|𝑃̂𝑋̂|𝑛〉 is non-zero when

(A) 𝑚 = 𝑛 ± 2 only

(B) 𝑚 = 𝑛 or 𝑚 = 𝑛 ± 2

(C) 𝑚 = 𝑛 ± 3 only

(D) 𝑚 = 𝑛 ± 1 only

Page 26 of 39

Q.44 A two-level quantum system has energy eigenvalues 𝐸1 and 𝐸2 . A perturbing
potential 𝐻 ′ = 𝜆Δ𝜎𝑥 is introduced, where Δ is a constant having dimensions of
0 1
energy, 𝜆 is a small dimensionless parameter, and 𝜎𝑥 = ( ). The magnitudes of
1 0
′
the first and the second order corrections to 𝐸1 due to 𝐻 , respectively, are

(A) 𝜆2 Δ2
0 and
|𝐸1 −𝐸2 |

(B) |𝜆Δ| 𝜆2 Δ2
and
2 |𝐸1 −𝐸2 |

(C) 𝜆2 Δ2
|𝜆Δ| and
|𝐸1 −𝐸2 |

(D) 1 𝜆2 Δ2
0 and
2 |𝐸1 −𝐸2 |

Page 27 of 39

Q.45 1
An electron with mass 𝑚 and charge 𝑞 is in the spin up state ( ) at time 𝑡 = 0. A
0
constant magnetic field is applied along the 𝑦-axis, 𝐵 = 𝐵0 𝑗̂, where 𝐵0 is a
𝑞𝐵0
constant. The Hamiltonian of the system is 𝐻 = − ℏ𝜔𝜎𝑦 , where 𝜔 = > 0 and
2𝑚
0 −𝑖
𝜎𝑦 = ( ). The minimum time after which the electron will be in the spin down
𝑖 0
1 1
state along the 𝑥-axis, i.e., ( ) , is
√2 −1

(A) 𝜋
8𝜔

(B) 𝜋
4𝜔

(C) 𝜋
2𝜔

(D) 𝜋
𝜔

Page 28 of 39

Q.46 1
A system of three non-identical spin particles has the Hamiltonian
2
𝐴
𝐻 = ℏ2 ( 𝑆1 + 𝑆2 ) ⋅ 𝑆3, where 𝑆1, 𝑆2 and 𝑆3 are the spin operators of particles
labelled 1, 2 and 3 respectively and 𝐴 is a constant with appropriate dimensions.
The set of possible energy eigenvalues of the system is

(A) 𝐴
0, , −𝐴
2

(B) 𝐴 𝐴
0, , −
2 2

(C) 3𝐴 𝐴
0, ,−
2 2

(D) 3𝐴 𝐴
0, − ,
2 2

Q.47 Which of the following option(s) is/are correct for a Type I superconductor?

(A) The phase transition to the normal state in the absence of a magnetic field is of
second order

(B) With increase in temperature, the critical magnetic field decreases linearly to zero

(C) Below the critical temperature, the entropy in the superconducting state is less than
that in the normal state

(D) The phase transition to the normal state in the presence of a magnetic field is of first
order

Page 29 of 39

Q.48 Consider two hypothetical nuclei 𝑋1 and 𝑋2 undergoing 𝛽 decay, resulting in nuclei
𝑌1 and 𝑌2 , respectively. The decay scheme and the corresponding 𝐽𝑃 values of the
nuclei are given in the figure. Which of the following option(s) is/are correct? (𝐽 is
the total angular momentum and 𝑃 is parity)

(A) 𝑋1 → 𝑌1 is Fermi transition and 𝑋2 → 𝑌2 is Fermi transition

(B) 𝑋1 → 𝑌1 is Fermi transition and 𝑋2 → 𝑌2 is Gamow-Teller transition

(C) 𝑋1 → 𝑌1 is Gamow-Teller transition and 𝑋2 → 𝑌2 is Fermi transition

(D) 𝑋1 → 𝑌1 is Gamow-Teller transition and 𝑋2 → 𝑌2 is Gamow-Teller transition

Page 30 of 39

Q.49 A point charge 𝑞 is placed at a distance 𝑑 above an infinite, grounded conducting
plate placed on the 𝑥𝑦 plane at 𝑧 = 0. The electrostatic potential in 𝑧 > 0 region is
1 𝑞
given by 𝜙 = 𝜙1 + 𝜙2 , where 𝜙1 = 2 2 2
and
4𝜋𝜖0 √𝑥 +𝑦 +(𝑧−𝑑)
1 𝑞
𝜙2 = − . Which of the following option(s) is/are correct?
4𝜋𝜖0 √𝑥 2 +𝑦 2 +(𝑧+𝑑)2

(A) 1 𝑞
The magnitude of the force experienced by the point charge 𝑞 is
16𝜋𝜖0 𝑑 2

(B) 1 𝑞2
The electrostatic energy of the system is
8𝜋𝜖0 𝑑

(C) 1
The induced surface charge density on the plate is proportional to
√𝑥 2 +𝑦 2 +𝑑 2

(D) The electrostatic potential 𝜙1 satisfies Poisson’s equation for 𝑧 > 0

Q.50 In coordinates (𝑡 , 𝑥), a contravariant second rank tensor 𝐴 has non-zero diagonal
components 𝐴𝑡𝑡 = 𝑃 and 𝐴𝑥𝑥 = 𝑄, with all other components vanishing, and 𝑃, 𝑄
being real constants. Here, 𝑡 is time and 𝑥 is space coordinate. Consider a Lorentz
transformation (𝑡, 𝑥) → (𝑡 ′ , 𝑥 ′ ) to another frame that moves with relative speed 𝑣 in
𝑡𝑡 𝑥𝑥
the +𝑥 direction, so that 𝐴 → 𝐴′ . If 𝐴′ and 𝐴′ are the diagonal components of
1
𝐴′ , then setting the speed of light 𝑐 = 1, and with 𝛾 = , which of the
√1−𝑣 2
following option(s) is/are correct?

(A) 𝐴′𝑡𝑡 = 𝛾 2 𝑃 + 𝛾 2 𝑣 2 𝑄

(B) 𝐴′𝑡𝑡 = 𝛾 2 𝑣 2 𝑃 + 𝑣 2 𝑄

(C) 𝐴′𝑥𝑥 = 𝛾 2 𝑣 2 𝑃 + 𝛾 2 𝑄

(D) 𝐴′𝑥𝑥 = 𝑣 2 𝑃 + 𝛾 2 𝑄

Page 31 of 39

Q.51 The Lagrangian of a particle of mass 𝑚 and charge 𝑞 moving in a uniform magnetic
field of magnitude 2𝐵 that points in the 𝑧 direction, is given by
𝑚
𝐿 = 2 𝑣 2 + 𝑞𝐵(𝑥𝑣𝑦 − 𝑦𝑣𝑥 ) where 𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 are the components of its velocity 𝑣.
If 𝑝𝑥 , 𝑝𝑦 , 𝑝𝑧 denote the conjugate momenta in the 𝑥, 𝑦, 𝑧 directions and 𝐻 is the
Hamiltonian, which of the following option(s) is/are correct ?

(A) 𝑑𝑥 1
= (𝑝 − 𝑞𝐵𝑦)
𝑑𝑡 𝑚 𝑥

(B) 𝑑𝑝𝑥 𝑞𝐵
= (𝑝 − 𝑞𝐵𝑥)
𝑑𝑡 𝑚 𝑦

(C) 𝑑𝑝𝑦 𝑞𝐵
= − (𝑝 + 𝑞𝐵𝑦)
𝑑𝑡 𝑚 𝑥

(D) 1 2
𝐻= [(𝑝𝑥 + 𝑞𝐵𝑦)2 + (𝑝𝑦 − 𝑞𝐵𝑥) + 𝑝𝑧2 ]
2𝑚

Q.52 A bead is constrained to move along a long, massless, frictionless horizontal rod
parallel to the 𝑥 axis. The rod itself is moving vertically upward along the
𝑧 direction against gravity with a constant speed, starting from 𝑧 = 0 at 𝑡 = 0, and
remains horizontal. The conjugate momenta are denoted by 𝑝𝑥 , 𝑝𝑦 , 𝑝𝑧 and the
Hamiltonian by 𝐻. Which of the following option(s) is/are correct?

(A) 𝐻 is the total energy of the system and is conserved

(B) 𝐻 is the total energy of the system and is not conserved

(C) 𝐻 is not the total energy of the system, but it is conserved

(D) 𝐻 is not the total energy of the system and is not conserved

Page 32 of 39

Q.53 In a one-dimensional Hamiltonian system with position 𝑞 and momentum 𝑝,
1
consider the canonical transformation (𝑞 , 𝑝) → (𝑄 = , 𝑃 = 𝑞 𝑝2 ), where 𝑄
𝑝
and 𝑃 are the new position and momentum, respectively. Which of the following
option(s) regarding the generating function 𝐹 is/are correct?

(A) 𝑞
𝐹 = 𝐹1 (𝑞, 𝑄) =
𝑄

(B) 𝐹 = 𝐹2 (𝑞, 𝑃) = √𝑃𝑞

(C) 𝑝
𝐹 = 𝐹3 (𝑝, 𝑄) = 2
𝑄

(D) 𝑃
𝐹 = 𝐹4 (𝑝, 𝑃) =
𝑝

Page 33 of 39

Q.54 The energy of a free, relativistic particle of rest mass 𝑚 moving along the 𝑥 axis in
one dimension, is denoted by 𝑇. When moving in a given potential 𝑉(𝑥), its
Hamiltonian is 𝐻 = 𝑇 + 𝑉(𝑥). In the presence of this potential, its speed is 𝑣,
conjugate momentum 𝑝, and the Lagrangian 𝐿. Then, which of the following
option(s) is/are correct?

(A)
𝑝2
𝐻 = 𝑐 2 √𝑚 2 + + 𝑉(𝑥)
𝑐2

(B) 𝑝𝑐
𝑣 =
√𝑝2 + 𝑚2 𝑐 2

(C)
𝑣2
𝐿 = 𝑚𝑐 2 √1 − − 𝑉(𝑥)
𝑐2

(D)
𝑣2
𝐿 = − 𝑚𝑐 2 √1 − − 𝑉(𝑥)
𝑐2

Page 34 of 39

Q.55 Consider the integral

1 𝑧4 − 1
𝐼= ∮ 𝑑𝑧,
2𝜋𝑖 (𝑧 − 𝑎 ) (𝑧 − 𝑏 )
𝑏 𝑎
where 𝑧 is a complex variable and 𝑎, 𝑏 are positive real numbers. The integral is
taken over a unit circle with center at the origin. Which of the following option(s)
is/are correct?

(A) 5
𝐼= when 𝑎 = 1, 𝑏 = 2
8

(B) 10
𝐼= when 𝑎 = 1, 𝑏 = 3
3

(C) 5
𝐼= when 𝑎 = 2, 𝑏 = 1
8

(D) 5
𝐼= when 𝑎 = 3, 𝑏 = 2
8

Q.56 A neutral conducting sphere of radius 𝑅 is placed in a uniform electric field of
magnitude 𝐸0 , that points along the 𝑧 axis. The electrostatic potential at any point 𝑟
𝑅3
outside the sphere is given by 𝑉(𝑟, 𝜃) = 𝑉0 − 𝐸0 𝑟 (1 − 𝑟 3 ) cos 𝜃, where 𝑉0 is the
constant potential of the sphere. Which of the following option(s) is/are correct?

(A) The induced surface charge density on the sphere is proportional to sin 𝜃

(B) As 𝑟 → ∞, 𝐸⃗ = 𝐸0 cos 𝜃 𝑟̂

(C) The electric field at any point is curl free for 𝑟 > 𝑅

(D) The electric field at any point is divergence free for 𝑟 > 𝑅

Page 35 of 39

Q.57 A point charge 𝑞 is placed at the origin, inside a linear dielectric medium of infinite
extent, having relative permittivity 𝜖𝑟 . Which of the following option(s) is/are
correct?

(A) 1
The magnitude of the polarization varies as
𝑟2

(B) 1
The magnitude of the polarization varies as
𝑟3

(C) The magnitude of the screened charge due to the dielectric medium is less than the
magnitude of the point charge 𝑞 for 𝜖𝑟 > 1

(D) The magnitude of the screened charge due to the dielectric medium is more than the
magnitude of the point charge 𝑞 for 𝜖𝑟 = 1

Q.58 A linear magnetic material in the form of a cylinder of radius 𝑅 and length 𝐿 is
placed with its axis parallel to the 𝑧 axis. The cylinder has uniform magnetization
𝑀𝑘̂. Which of the following option(s) is/are correct?

(A) The magnetic field at any point outside the cylinder can be expressed as the gradient
of a scalar function

(B) The bound volume current density is zero

(C) The surface current density on the curved surface is non-zero

(D) The surface current densities on the flat surfaces (top and bottom) are non-zero

Page 36 of 39

Q.59 Cyclotrons are used to accelerate ions like deuterons (𝑑) and 𝛼 particles. Keeping
the magnetic field same for both, 𝑑 and 𝛼 are extracted with energies 10 MeV and
20 MeV with extraction radii 𝑟𝑑 and 𝑟𝛼 , respectively. Taking the masses
𝑟𝛼
𝑀𝑑 = 2000 MeV/𝑐 2 and 𝑀𝛼 = 4000 MeV/𝑐 2 , the value of (in integer) is
𝑟𝑑
______

Q.60 In the transistor circuit shown in the figure, 𝑉𝐵𝐸 = 0.7 V and 𝛽𝐷𝐶 = 400. The value
of the base current in 𝜇A (rounded off to one decimal place) is ___________

Q.61 Consider the set {1, 𝑥, 𝑥 2 }. An orthonormal basis in 𝑥 ∈ [−1,1] is formed from
these three terms, where the normalization of a function 𝑓(𝑥) is defined via
1
∫−1 𝑥 2 [𝑓(𝑥)]2 𝑑𝑥 = 1. If the orthonormal basis set is
3 5 1 21
(√ , √ 𝑥 , √ (5𝑥 2 − 3)) , then the value of 𝑁 (in integer) is ______
2 2 2 𝑁

Page 37 of 39

The Hamiltonian for a one dimensional system with mass 𝑚, position 𝑞 and
Q.62 𝑝2
momentum 𝑝 is 𝐻(𝑝, 𝑞) = + 𝑞 2 𝐴(𝑞), where 𝐴(𝑞) is a real function of 𝑞. If
2𝑚
𝑑2𝑞 𝑑𝐴(𝑞) 𝐴(𝑞)
𝑚 = −5𝑞𝐴(𝑞), then =𝑛 . The value of 𝑛 (in integer) is ______
𝑑𝑡 2 𝑑𝑞 𝑞

3
A system of five identical, non-interacting particles with mass 𝑚 and spin is
Q.63 2
confined to a one-dimensional potential well of length 𝐿. If the lowest energy of the
𝜋 2 ℏ2
system is 𝑁 , the value of 𝑁 (in integer) is _____
2𝑚𝐿2

Q.64 A wheel of mass 4𝑀 and radius 𝑅 is made of a thin uniform distribution of mass
3𝑀 at the rim and a point mass 𝑀 at the center. The spokes of the wheel are
massless. The center of mass of the wheel is connected to a horizontal massless rod
of length 2𝑅, with one end fixed at O, as shown in the figure. The wheel rolls
⃗ is the total angular
without slipping on horizontal ground with angular speed Ω. If 𝐿
⃗
𝑑𝐿
momentum of the wheel about O, then the magnitude | | = 𝑁(𝑀𝑅2 Ω2 ). The
𝑑𝑡
value of 𝑁 (in integer) is ____

Page 38 of 39

The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop.
Q.65 The opamp runs from ±15 V supplies. If a +1V signal is applied at the input, the
output voltage (rounded off to one decimal place) is ____

Page 39 of 39

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