GATE 2026 Question Paper Physics (PH)

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GATE 2026
Question Paper ·
EXAM YEAR TYPE SUBJECT

GATE 2026 Question Paper Physics (PH)

Notes · Sample Papers · Previous Year Papers · Mock Tests

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

Q.1 “He often _____ the numbers. False claims are not going to help. Honesty _____
trust”, said the manager.
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Choose the option with the correct order of words to fill the blanks.
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(A) a exaggerates; engenders

(B) excels; encourages

(C) aggravates; alleviates

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(D) diminishes; eliminates

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Q.2 In the sequence of tiles shown below, the missing tile indicated by the question
mark should be

(A)

(B)

(C)

(D)

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Q.3 A school has 100 students distributed among 1st to 10th standards.

Based on this, which one of the following statements is always correct?

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There are at least 10 students who belong to the same standard.
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(B) a There is at least one student in each standard.

(C) There are at most 10 students in 10th standard.

(D) The total number of students from 1st to 5th standards is at least 50.

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Q.4 How many 3-digit numbers can be formed using three distinct single digit prime
numbers?
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(A) 64

(B) 24
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Q.5 In a group of students, 10 students like Mathematics, 12 students like English,
4 students like both Mathematics and English, and 6 students like neither
Mathematics nor English. The number of students in the group is ____

(A) 18

(B) 20

(C) 24

(D) 32

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Q.6 – Q.10 Carry TWO marks Each

Q.6 Charity : P :: Retaliation : Q

Choose the appropriate pair of words P and Q that fit the analogy.

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P = Parsimonious; Q = Vengeful
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(B) a P = Altruistic; Q = Amicable

(C) P = Resentful; Q = Spiteful

(D) P = Magnanimous; Q = Vindictive

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Q.7 A paper shown in Panel I is folded along the dashed lines (- - -) to construct a
cube. The shaded regions shown in Panel I appear on the outer surface of the cube.
Referring to cubes shown in Panel II, which one of the options is correct?

(A) Only (i) can correspond to the unfolded cube in Panel I.

(B) Only (ii) can correspond to the unfolded cube in Panel I.

(C) Both (i) and (ii) can correspond to the unfolded cube in Panel I.

(D) Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.

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Q.8 Consider the cube shown below with its 8 corners labelled a, b, c, d, e, f, g,
and h. The figure is representative. All corners are to be colored such that any two
corners that are connected by an edge must be of different colors. The minimum
number of colors required to achieve this is ________

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(A) 8
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(B) 4

(C) 3

(D) 2
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Q.9 Four hills H1, H2, H3, and H4 are present in an area. The following observations
are made about them:

i. Neither H2 nor H3 is the easternmost hill.
ii. Neither H2 nor H3 is the westernmost hill.
iii. Neither the easternmost hill nor the westernmost hill is the southernmost
hill.
iv. Two hills are located to the west of H2.
v. The southernmost hill has at least two hills to its east.

The southernmost hill is ________.

(A) H1

(B) H2

(C) H3

(D) H4

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Q.10 As shown in the figure, circle 𝐶1 with center 𝑂1 and radius 𝑟1 touches the square
𝑉𝑊𝑋𝑌 at points 𝑃 and 𝑄 while circle 𝐶2 with center 𝑂2 and radius 𝑟2 touches the
square 𝑉𝑊𝑋𝑌 at points 𝑅 and 𝑆. The two circles touch each other at 𝑇.

̅̅̅̅ = 𝑉𝑊
Given 𝑟1 = 1 cm and 𝑉𝑌 ̅̅̅̅̅ = 4 cm, 𝑟2 = _____ cm.

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(A)
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a (C) 7 − 4√2

(D) 5 + 3√2

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Q.11 – Q.35 Carry ONE mark Each

In free space, an electromagnetic wave is travelling whose wavevector is
Q.11
𝑘⃗ = 10(𝑥̂ + √3𝑦̂) m−1 . The electric field component of this electromagnetic wave is
given by 𝐸⃗ (𝑟, 𝑡) = 𝑧̂ 600 cos(𝑘⃗ ⋅ 𝑟 − 𝜔𝑡) V.m−1 . The speed of light in free space is
𝑐 = 3.0 × 108 m.s −1. The corresponding magnetic field 𝐵⃗ (𝑟, 𝑡) is

(A) ⃗ (𝑟, 𝑡) = 2 × 10−6 (√3𝑥̂ − 𝑦̂) cos(𝑘⃗ ⋅ 𝑟 − 𝜔𝑡) V.m−2 .s
𝐵

(B) ⃗ (𝑟, 𝑡) = 10−6 (√3𝑥̂ − 𝑦̂) cos(𝑘⃗ ⋅ 𝑟 − 𝜔𝑡) V.m−2 .s
𝐵

(C) ⃗ (𝑟, 𝑡) = 2 × 10−5 (−√3𝑥̂ + 𝑦̂) cos(𝑘⃗ ⋅ 𝑟 − 𝜔𝑡) V.m−2 .s
𝐵

(D) ⃗ (𝑟, 𝑡) = 10−5 (√3𝑥̂ − 𝑦̂) cos(𝑘⃗ ⋅ 𝑟 − 𝜔𝑡) V.m−2 .s
𝐵

An infinitely large non-conducting thin sheet in the 𝑥𝑦 plane (𝑧=0) carries a uniform
Q.12
surface charge density 𝜎 = 17.70 × 10−12 C.m−2. The electric field in the region 𝑧 < 0
is 𝐸⃗2 = 𝑥̂ + 2𝑦̂ + 3𝑧̂ . Then, the electric field 𝐸⃗1 in the region 𝑧 > 0 will be

(𝜖0 = 8.85 × 10−12 C 2 . N −1 . m−2)

(A) 𝐸⃗1 = 𝑥̂ + 2𝑦̂ + 5𝑧̂

(B) 𝐸⃗1 = 𝑥̂ + 2𝑦̂ + 4𝑧̂

(C) 𝐸⃗1 = 𝑥̂ + 2𝑦̂ + 3𝑧̂

(D) 𝐸⃗1 = 𝑥̂ + 4𝑦̂ + 𝑧̂

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Q.13 Consider an operator 𝐴̂ which is not Hermitian. Find the possible values of 𝑐 and 𝑑 such
that the operator (𝑐 𝐴̂ – 𝑑 𝐴̂† ) is Hermitian.

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(A) 𝑐 = 𝑖 and 𝑑 = 𝑖
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𝑐 = 1 and 𝑑 = 1
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(C) a 𝑐 = −1 and 𝑑 = 𝑖

(D) 𝑐 = 𝑖 and 𝑑 = −𝑖

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Q.14 ⃗ × (𝐴 𝜓) is equivalent to the expression
For a scalar field 𝜓(𝑟) and a vector field 𝐴(𝑟), ∇

(A) ⃗ × 𝐴) − 𝐴 × (∇
𝜓(∇ ⃗ 𝜓)

(B) ⃗ × 𝐴) + 𝐴 × (∇
𝜓(∇ ⃗ 𝜓)

(C) Null vector

(D) ⃗ 𝜓) − 𝜓(∇
𝐴 × (∇ ⃗ × 𝐴)

Which of the following options is correct for transformation of electric field 𝐸⃗ and
Q.15
⃗ under time reversal, i.e., 𝑡 → −𝑡 ?
magnetic field 𝐵

(A) 𝐸⃗ → 𝐸⃗ and 𝐵
⃗ →𝐵
⃗

(B) 𝐸⃗ → −𝐸⃗ and 𝐵
⃗ →𝐵
⃗

(C) 𝐸⃗ → 𝐸⃗ and 𝐵
⃗ → −𝐵
⃗

(D) 𝐸⃗ → −𝐸⃗ and 𝐵
⃗ → −𝐵
⃗

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Q.16 On a horizontal plane, a projectile of mass 𝑚 is launched from the ground with speed 𝑣0
at an angle 𝜃0 with the horizontal. In addition to the gravitational force (𝑚𝑔), it also
experiences a drag force 𝐹𝑑𝑟𝑎𝑔 = −𝛾𝑣 , where 𝑣 is its velocity and 𝛾 is a constant. It hits
the ground at a distance 𝑅 from the point of launch with its velocity making an angle
𝜃 with the horizontal, as shown schematically in the figure. Then which of the following
options is correct?
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(A) 𝑣02 𝑠𝑖𝑛2𝜃
𝑅= 𝜃 < 𝜃0
𝑔

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(B) 𝑅<
𝑣02 𝑠𝑖𝑛2𝜃0
𝜃 < 𝜃0
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𝑔
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𝑣02 𝑠𝑖𝑛2𝜃0
g la
(C) 𝑅<
𝑔
𝜃>𝜃
a 0

(D) 𝑣02 𝑠𝑖𝑛2𝜃
𝑅= 𝜃 > 𝜃0
𝑔

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Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and
Q.17
two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram
for the cycle is

(A)

(B)

(C)

(D)

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The formula for energy 𝐸 of a photon gas at temperature 𝑇 in a two-dimensional box at
Q.18
equilibrium with 𝑔2𝑑 (𝜈) denoting the density of states of photons is given below where
symbols 𝜈, ℎ and 𝑘𝐵 have their standard meaning. The specific heat (𝐶𝑉 ) of this photon
gas obeys
∞
ℎ𝜈
𝐸 = ∫ 𝑑𝜈 𝑔2𝑑 (𝜈)
ℎ𝜈
exp ( )−1
m
0
𝑘𝐵 𝑇
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(A)
g a
𝐶l ∝ 𝑇 ag
a 𝑉

(B) 𝐶𝑉 ∝ 𝑇 2

(C) 𝐶𝑉 ∝ 𝑇 3

(D) 𝐶𝑉 ∝ 𝑇 4
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Q.19
a
For the electric field of an electromagnetic
statements is correct?
𝜋
𝐸⃗ = 𝑥̂ 𝐸0 𝑐𝑜𝑠(𝜔𝑡) + 𝑦̂ 2𝐸0 𝑐𝑜𝑠 (𝜔𝑡 + )
2

(A) The electric field is linearly polarised with slope 2.
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The electric field is circularly polarised with radius 𝐸0 .
s em
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to minor axis being 2.
a
(D) The electric field is unpolarised with the two components being phase shifted by 𝜋/2.

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A gas of non-interacting 4He atoms (of mass 𝑚) is in a three-dimensional trap whose
Q.20
energy levels can be approximated by those of a harmonic oscillator potential
1
𝑉(𝑥, 𝑦, 𝑧) = 𝑚𝜔2 (𝑥 2 + 𝑦 2 + 𝑧 2 ). The chemical potential of the gas at 𝑇=0 K is
2

(A) 0

1
(B) ℏ𝜔
2

(C) 3
ℏ𝜔
2

(D) 3ℏ𝜔

1 1 1 1
Q.21 Given |𝑣1 ⟩ = ( ) and |𝑣2 ⟩ = ( ), the tensor product |𝑣1 ⟩ ⊗ |𝑣2 ⟩ is
√2 𝑖 √2 −𝑖

1
(A) 1 −𝑖
( )
2 𝑖
1
1
1
(B) (𝑖 )
2 −𝑖
1
1 1 𝑖
(C) ( )
2 𝑖 −1
1 1 𝑖
(D) ( )
2 −𝑖 1

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Sketch of a two-dimensional vector field 𝑉 ⃗ is shown below. Here, length and arrow head
Q.22
of the arrows denote magnitude and direction of the vector field, respectively. Which of
the following statements is correct for ⃗∇ × 𝑉⃗ ?

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(A) It is zero everywhere in the two-dimensional space.

(B) Its magnitude is non-zero and its direction is out of the two-dimensional plane.
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(C)
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Its magnitude is non-zero and its direction is into the two-dimensional plane.
s
(D) gla and below the x-axis.
It points in opposite directionsaabove

Q.23 Which one of the following is an allowed process?

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𝜋− + 𝑝 → 𝜋0 + 𝑛
s e
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a
(C) 𝑝 + 𝑝̅ → Λ0 + Λ0

(D) 𝑝 + 𝑝̅ → 𝛾

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Q.24 Given 𝑄 is the electromagnetic charge and 𝑆 is the strangeness quantum number,

identify the particle(s) for which (𝑄 − 𝑆) = 0 is satisfied.

(A) Σ∗ −

(B) 𝐾+

(C) Ω−

(D) Δ++

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Schematic variation of the specific heat 𝐶𝑝 of an ideal gas of diatomic molecules with
Q.25
temperature 𝑇 is shown in the figure below. For rotational energy 𝐸𝑅 and vibrational
energy 𝐸𝑣 of the molecule, which of the following options is/are correct? Here 𝑘𝐵 is the
Boltzmann constant.

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(A) 𝐸𝑅 ≅ 𝑘𝐵 𝑇1

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(B) 𝐸𝑅 ≅ 𝑘𝐵 𝑇2
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(C) 𝐸𝑣 ≅ 𝑘𝐵 𝑇1
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a
(D) 𝐸𝑣 ≅ 𝑘𝐵 𝑇2

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Q.26 If the perturbation 𝑉 = 𝜆𝑥 3 is added to the Hamiltonian of a one-dimensional harmonic
oscillator, the matrix element ⟨𝑚|𝑉|0⟩ is/are non-zero for which of the following states?
Here, the eigenstates of the harmonic oscillator are denoted by |𝑛⟩.

(A) |𝑚 = 3⟩

(B) |𝑚 = 1⟩

(C) |𝑚 = 2⟩

(D) |𝑚 = 5⟩

Q.27 For which of the following functions does the Laplacian vanish?

(A) 𝑥𝑒 𝑦 − 𝑦𝑒 𝑥

(B) 𝑥 cos(𝑦) − 𝑦 cos(𝑥)

(C) 𝑒 𝑥+𝑖𝑦

(D) 𝑦3
𝑦𝑥 2 − − 𝑥𝑦
3

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A projectile of mass 𝑚 is launched from the ground with the initial speed 𝑣0 at an angle
Q.28
30° from the horizontal. Take the ground to be horizontal. Ignoring the drag, the
magnitude of Hamilton’s action ∫ 𝐿𝑑𝑡 for the particle from the beginning till it hits the
𝑚𝑣 3
ground is 𝑓 × ( 𝑔 0 ). The value of 𝑓 (rounded off to two decimal places) is _____

Consider an electron in the energy eigenstate 𝜓211 (𝑟) of the hydrogen atom. Given that
Q.29
m
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the radial probability distribution of the electron in such a state takes its maximum value
m
at 𝑟 = 𝑛0 𝑎, where 𝑎 is the Bohr radius, and 𝑛0 is an integer. The value of 𝑛0 (in integer)
is _____
.co e m
s empart of the wavefunction 𝜓 (𝑟) is given by 𝑅 (𝑟) = 𝑟𝑒 . glas
The radial
1 −𝑟/2𝑎

a
211 21

a
√24𝑎5

glA dielectric
Q.30 a
−2
sphere carries a uniform polarization 𝑃 = 26 μC. cm . The magnitude of the
9 −1
electric field at the center of the sphere is 𝐸 × 10 N. C . The value of 𝐸 (rounded off to
one decimal place) is _____
(𝜖0 = 8.85 × 10−12 C 2 . N −1 . m−2 )

Consider a metal-superconductor junction connected to a dc voltage 𝑉. At 𝑇 < 𝑇𝑐 , where
Q.31
𝑇𝑐 is the superconductor’s transition temperature, the current 𝐼 versus 𝑉 behavior of this
junction is shown schematically in the figure below. If the superconducting energy gap is

m
𝐷 meV. The value of 𝐷 (rounded off to one decimal place) is _____

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For the energy dispersion of an electron in a one-dimensional solid
Q.32
𝐸(𝑘) = 𝐸0 − 2𝛾 cos(𝑘𝑎), the ratio of the effective mass of the electron in the solid to the
free electron mass (𝑚𝑒 ) at 𝑘 = 0 is 𝑅0 . Taking 𝛾 = 0.5 eV and 𝑎 = 0.5 nm, the value of 𝑅0
(rounded off to two decimal place) is _____
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m c
kg, electron charge = 1.6 × 10 . C)
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−34 −31 −19

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(ℏ = 1.054 × 10 J. s, 𝑚𝑒 = 9.1 × 10

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The specific heat 𝐶𝑝 (𝑇) of one mole of a material as a function of temperature 𝑇 is given
Q.33
as 𝐶𝑝 (𝑇) = 𝐴𝑇 + 𝐵𝑇 3, where 𝐴 = 0.695 mJ. mol−1 . K −2 and 𝐵 = 0.045 mJ. mol−1 . K −4.
When 𝑇 is changed from 1 K to 10 K at constant pressure, then the change in entropy
Δ𝑆 in mJ. mol−1 . K −1 (rounded off to one decimal place) is _____

Raman spectrum of a molecule was recorded using a source of wavelength 5000 Å. The
Q.34
first Stokes line is observed at 5100 Å. The first anti-Stokes line will appear at a
wavelength 𝐿 (in Å). The value of 𝐿 (rounded off to nearest integer) is _____

Q.35 A rocket of length 18.0 m is moving at speed 0.9c (where c is the speed of light) parallel
to its own length, relative to the earth. The length of the rocket measured in meters by an
observer on earth (rounded off to two decimal places) is _____

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Q.36 – Q.65 Carry TWO marks Each

Q.36 The function 𝑓(𝑧) of complex variable 𝑧 given below,

𝑧 2 − 5𝑧 + 4
𝑓(𝑧) =
𝑧 3 + 4𝑧 − 𝑧 2 − 4
m
.co
has singular points at 𝑧 =
m
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s em l a
g l a ag
(A) a 1 and (2 − 𝑖)

(B) 2𝑖 and −2𝑖

(C) 1 and (2 + 𝑖)

(D) (2 + 𝑖) only
m
.co
s em
g la0 1
Consider the Pauli matrices 𝜎a= (
0 −𝑖 1 0
Q.37 𝑥),𝜎 = ( ),𝜎 = (
𝑦 ). 𝑧
1 0 𝑖 0 0 −1

The value of Tr(𝜎𝑧 [ 𝜎𝑥 , 𝜎𝑦 ]) is

(A) 2𝑖
m
m
c. o(B) m.co
m
𝑖
s e
s e g l a
g la (C) 4𝑖 a
a 𝑖
(D) 2

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la
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Q.38 Which one of the following statements is true?

(A) In the decay 𝜇 + → 𝑒 + + 𝜈𝑒 + 𝜈̅𝜇 , CPT is violated.

(B) The decay Λ → 𝑝+ + 𝜋 − is allowed and strangeness is violated.

(C) The decay 𝑝+ → 𝑒 + + 𝛾 is allowed.

(D) The decay Ω− → Ξ 0 + 𝐾 − is allowed.

The Hamiltonian for a quantum particle of mass 𝑚 is given below, where 𝜔 < Ω. The
Q.39
Schrö dinger equation for this system can be solved exactly using the orthogonal
𝑥 −𝑥 𝑥 +𝑥
transformations: 𝑥 = 1 2 and 𝑦 = 1 2 .
√2 √2

ℏ2 𝜕 2 𝜕2 1
𝐻= − [ 2 + 2 ] + 𝑚Ω2 (𝑥 2 + 𝑦 2 ) + 𝑚𝜔2 𝑥𝑦
2𝑚 𝜕𝑥 𝜕𝑦 2

The ground state energy of this system is

(A) ℏ
[√Ω2 − 𝜔 2 + √Ω2 + 𝜔 2 ]
2

(B) ℏ [√Ω2 − 𝜔 2 + √Ω2 + 𝜔 2 ]

ℏ
(C) [√Ω2 + Ω𝜔 + √Ω2 − Ω𝜔 ]
2

(D) ℏ [√Ω2 + Ω𝜔 − √Ω2 − Ω𝜔 ]

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Consider two particles with angular momenta 𝑗1 = 2ℏ and 𝑗2 = ℏ/2. If the expression
Q.40
𝑐 |𝑗 = 2, 𝑚1 = 1⟩|𝑗2 = 1/2, 𝑚2 = 1/2⟩ +
|𝑗 = 5/2,𝑚 = 3/2⟩ = { 1 1
𝑐2 |𝑗1 = 2, 𝑚1 = 2⟩|𝑗2 = 1/2, 𝑚2 = −1/2⟩
gives an eigenstate of the total angular momentum of the two particles, using standard
notation. Which of the following is true?
(Hint: 𝐽̂± |𝑗, 𝑚⟩ = √𝑗(𝑗 + 1) − 𝑚 (𝑚 ± 1) |𝑗, 𝑚 ± 1⟩)
m
m .co
.co s e m
s em l a
(A)
g𝑐 la
= , 𝑐 =
2 1
ag
a 1 √5 2 √5

1 2
(B) 𝑐1 = , 𝑐2 =
√5 √5
1 1
(C) 𝑐1 = , 𝑐2 =
√2 √2

(D) 𝑐1 = 0, 𝑐2 = 1
m
.co
s em
The energy 𝐸 and degeneracy 𝑑 ofla
Q.41 g the second excited state of a three-dimensional,
a with angular frequency 𝜔 are
isotropic quantum harmonic oscillator

(A) 7
𝐸= ℏ𝜔, 𝑑=6
2

m
.co
(B) 7
m 𝐸= ℏ𝜔, 𝑑=3

.co
2
e m
e m l as
las (C) 𝐸=
5
2
ℏ𝜔, 𝑑=3
ag
ag
(D) 5
𝐸= ℏ𝜔, 𝑑=6
2

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Q.42 Two identical particles with a fixed total energy 𝐸 = 2ℏ𝜔 are in thermal equilibrium in a
1
one-dimensional harmonic oscillator potential 2 𝑚𝜔2 𝑥 2 . Let the entropy of the particles
1 ℏ
be denoted by 𝑆𝐹 if they are fermions with spin
2
(𝑠𝑧 = ± 2 ) and 𝑆𝐵 if they are
bosons with spin 0. Then, which of the following options is correct?
(𝑘𝐵 is the Boltzmann constant)

(A) 𝑆𝐹 = 𝑘𝐵 ln 2, 𝑆𝐵 = 𝑘𝐵 ln 2

(B) 𝑆𝐹 = 2𝑘𝐵 ln 2, 𝑆𝐵 = 0

(C) 𝑆𝐹 = 4𝑘𝐵 ln 2, 𝑆𝐵 = 0

(D) 𝑆𝐹 = 2𝑘𝐵 ln 2, 𝑆𝐵 = 𝑘𝐵 ln 2

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Q.43 In the circuit shown, V(t) = 2 𝑠𝑖𝑛(2000𝜋t) Volts, where t is in seconds, which of the
following options is correct?

(take the opamp to be ideal)

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a

o m
V (t) is square wave with peak-to-peakm
. c
(A) out
s e voltage = 30 V and time period is 1 ms.

g la voltage = 4 V and time period is 1 ms.
(B) out
a
V (t) is a sine wave with peak-to-peak

(C) Vout (t) is sine wave with peak-to-peak voltage = 30 V and time period of 1 ms.

(D) Vout (t) is square wave with peak-to-peak voltage = 4 V and time period is 1 ms.

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Q.44 Considering the circuit and the associated signals measured at different pins (numbered as
1, 2, 3, 4, 5) shown in the figure, the correct option is

(A) NOT gate between pins 1 and 2 is faulty.

(B) NOR gate is faulty.

(C) NOT gate between pins 4 and 5 is faulty.

(D) The NOR and output NOT gates, are both faulty.

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A gas of 𝑁 classical particles that can occupy energy levels, 𝜖1 and 𝜖2 = 𝜖1 + Δ is in
Q.45
equilibrium with a reservoir at temperature 𝑇. From the schematics shown below, choose
the correct dependence of the internal energy 𝑈 on 𝑇 .

m
(A)
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a
(B)

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s e
la
(C)

ag

(D)

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a

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The Lagrangian
Q.46 1 1
𝐿0 = 𝑚𝑞̇ 2 − 𝑚𝜔2 𝑞2
2 2
𝑑𝑓(𝑞)
with the generalized coordinate 𝑞 is transformed to 𝐿 = 𝐿0 + 𝛼 𝑑𝑡
. Consider the
following statements:
(i) Expression for the canonical momentum does not change.
(ii) The equation of the motion does not change.
Which of the following options is correct for the above statements?

(A) Both (i) and (ii) are correct.

(B) Both (i) and (ii) are not correct.

(C) (i) is correct and (ii) is not correct.

(D) (i) is not correct and (ii) is correct.

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Q.47
An infinitely large thin sheet in the 𝑥𝑦-plane carries uniform positive charge density and
is moving with constant velocity 𝑣 in the +𝑥 direction (see figure below). The direction
of the corresponding Poynting vector is

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a
(A) +𝑥 for both 𝑧 < 0 and 𝑧 > 0

(B) +𝑥 for 𝑧 < 0 and −𝑥 for 𝑧 > 0

m
(C) −𝑥 for 𝑧 < 0 and +𝑥 for 𝑧 > 0

m .co
s e
(D) −𝑥 for both 𝑧 < 0 and 𝑧 > 0
g la
a
A positive point charge is fixed at the origin. At some distance from it on the 𝑥 axis, a
Q.48
point dipole is kept pointing in the +𝑦 direction. The force on the dipole is

m
m
(A) 0
.co
m .co s e m
s e (B) in the +𝑦 direction
g l a
g la a
a (C) in the −𝑦 direction

(D) in the +𝑥 direction

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Q.49 Two frames 𝑆 (solid lines) and 𝑆′ (dashed lines) with common origin are shown in the
figure below. Frame 𝑆 is inertial while 𝑆′ is rotating about the common 𝑧-axis. There is a
point mass fixed at 𝑃 on the 𝑥-axis of the 𝑆 frame. The magnitude of the centrifugal force
and the Coriolis force experienced by the mass in the 𝑆′ frame is 𝐹𝑐𝑒𝑛 and 𝐹𝑐𝑜𝑟 ,
respectively. Which of the following options is correct for these forces?

(A) 𝐹𝑐𝑒𝑛 = 0 and 𝐹𝑐𝑜𝑟 = 0

𝐹
(B) 𝐹𝑐𝑒𝑛 ≠ 0 and 𝐹𝑐𝑜𝑟 ≠ 0 and 𝐹𝑐𝑒𝑛 = 𝑐𝑜𝑟
2

(C) 𝐹𝑐𝑒𝑛 ≠ 0 and 𝐹𝑐𝑜𝑟 ≠ 0 and 𝐹𝑐𝑒𝑛 = 2𝐹𝑐𝑜𝑟

(D) 𝐹𝑐𝑒𝑛 ≠ 0 and 𝐹𝑐𝑜𝑟 ≠ 0 and 𝐹𝑐𝑒𝑛 = 𝐹𝑐𝑜𝑟

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Q.50 Which of the following operators is/are self-adjoint?

𝑑2
(A) 𝑑
m
.co
𝑥2 + 3𝑥 + 𝑥2
m
𝑑𝑥 2 𝑑𝑥

c o m
𝑑. s e
(1 − 𝑥e)m − 2𝑥 a
(B) 2
𝑑
s
2
+ 3𝑥 l
la ag
𝑑𝑥 𝑑𝑥 2

ag 2
(C) 𝑑 𝑑
(3𝑥 − 4𝑥 3 ) + (3 − 12𝑥 2 ) + 12
𝑑𝑥 2 𝑑𝑥

(D) 𝑑2 𝑑 5𝑥
𝑥 2 + 𝑥2 +
𝑑𝑥 𝑑𝑥 3

m
m .co
Q.51
s e
Consider two operators 𝐴̂ and 𝐵̂ which are related as 𝐴̂ = exp(𝑖𝜃𝐵̂). If 𝜃 is a non-zero
la
real number, which of the following statements is/are true?
g
a

(A) If 𝐵̂ is Hermitian, then 𝐴̂ is unitary.

(B) If 𝐵̂ is anti-Hermitian, then 𝐴̂ is unitary.

m
m .co
.co
(C) If 𝐵̂ is Hermitian, then |𝐷𝑒𝑡(𝐴̂)| = 1.
e m
e m l as
las (D) If 𝐵̂ is anti-Hermitian, then 𝐴̂ is Hermitian.
ag
ag

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Q.52 Consider operators 𝐴̂, 𝐵̂, and 𝐶̂ for three observables of a quantum system satisfying
[𝐴̂, 𝐵̂] = 0, [𝐵̂, 𝐶̂ ] = 0, and [𝐴̂, 𝐶̂ ] ≠ 0, with uncertainties Δ𝐴, Δ𝐵, Δ𝐶, respectively. From
the options given below, which is/are implied by the commutation relations among
𝐴̂, 𝐵̂, and 𝐶̂ ?

(A) Δ𝐴 Δ𝐵 > 0

(B) Δ𝐴 Δ𝐶 > 0

(C) 𝐴̂, 𝐵̂ can be simultaneously diagonalized.

(D) 𝐴̂, 𝐵̂, 𝐶̂ can be simultaneously diagonalized.

Q.53 Consider the distribution of outcomes generated by 𝑁 (𝑁 ≫ 1) independent throws of
(i) a coin or (ii) a six-sided dice. A coin (dice) is unbiased if both (all) its sides have equal
probability to show up in a throw; it is biased otherwise. For the cases (i) and (ii) above,
which of the following statements is/are true?

(A) The entropy of an unbiased coin is smaller than that of an unbiased dice.

(B) The entropy of an unbiased coin is greater than that of an unbiased dice.

(C) The entropy of a biased dice is smaller than that of an unbiased dice.

(D) The entropy of a biased coin is greater than that of an unbiased coin.

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Q.54 The dispersion (𝐸(𝑘)) of the conduction band (CB) and valence band (VB) for a
semiconductor are shown schematically in the figure. Considering the possibility of an
electron making a transition from the bottom of the CB to the top of the VB, which of the
following options is/are correct?

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a

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(A) The transition is forbidden.

m .co
s e
(B)
g la
A photon can be emitted with an energy exactly equal to 𝐸𝑔 .

a
(C) A photon can be emitted with an energy less than 𝐸𝑔 .

(D) A phonon can be created with a crystal momentum ℏ𝑞.

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Q.55 A symmetric rigid body has moment of inertia 𝐼1 , 𝐼2 , 𝐼3 about its principal axes 1, 2, and
3, respectively, with 𝐼1 = 𝐼3 = 𝐼⊥ and 𝐼2 ≠ 𝐼⊥ . It is rotating in space with no torque on it
so that its angular momentum 𝐿⃗ is constant. Let 𝜔1 , 𝜔2 , 𝜔3 be the components of its
angular velocity along the principal axes 1, 2, and 3, respectively. Which of the following
quantities is/are constant during the motion of this rigid body?

(A) 𝜔1 + 𝜔3

(B) 𝜔12 + 𝜔32

(C) Angle between axis 2 and 𝐿⃗

(D) 𝜔2

Q.56 An 𝛼 particle moves towards a fixed nucleus carrying charge 𝑍𝑒, with initial speed 𝑣0 and
impact parameter 𝑏. Starting from a large distance from the nucleus, its distance of closest
approach is 𝑟𝑚 and its speed there is 𝑣𝑚 . Then which of the following options is/are
correct?
1 𝑍𝑒 2
(𝑘 = and 𝑟0 = 𝑘 )
4𝜋𝜖0 𝑚𝑣02

(A) 𝑣0 𝑏 = 𝑣𝑚 𝑟𝑚

(B) 𝑣0 𝑏 = 2𝑣𝑚 𝑟0

𝑏 𝑏2 𝑏
(C) For 𝑟 ≪ 1, 𝑟𝑚 = 4𝑟0 + 2𝑟 ignoring higher order corrections in 𝑟
0 0 0

𝑏 𝑏2 𝑏
(D) For 𝑟 ≪ 1, 𝑟𝑚 = 4𝑟0 + 8𝑟 ignoring higher order corrections in 𝑟
0 0 0

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Q.57 Consider a particle of mass 𝑚 = 9.0 × 10−5 kg and charge 𝑞 = 3.0 × 10−4 C in a
uniform electromagnetic field 𝐸⃗ = 2 𝑥̂ V. m−1 , 𝐵 ⃗ = 3 𝑧̂ V. m−2 . s. The particle is
released from the coordinates (0, 5 m , 0) at time 𝑡 = 0. Starting from initial speed zero, it
comes back to the 𝑦-axis for the first time at time 𝑡. The value of 𝑡 in seconds (rounded
off to two decimal places) is _____

m
.co
An atom has two energy levels with energy difference 2.2 eV between them. A gas of
Q.58
m
.co
these atoms is with 8 × 1020 atoms in the upper state and 5 × 1020 atoms in the lower
e m
state. Ignoring spontaneous emission, the maximum possible energy released by this gas
em
of atoms by stimulated emission is 𝐸 Joules. The value of 𝐸 (rounded off to one decimal
l as
la s
place) is _____
ag
ag(𝑒 = 1.6 × 10 −19
C)
𝐴
Q.59 The Hamiltonian of two interacting spin-1/2 particles is 𝐻 = 2 𝑆1 ⋅ 𝑆2 , where 𝑆1 and 𝑆2
ℏ
are the spin angular momenta of particles 1 and 2, respectively. Here, 𝐴 = 10.56 eV. The
energy in eV required to induce an excitation from the ground state to the excited state
(rounded off to two decimal places) is _____

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Q.60 The output signal (current 𝐼𝑑 ) of a reversed biased (with a voltage 𝑉𝑏 ) photodiode, on
which light is incident, is fed to an amplifier (see figure). The output voltage is digitized
by a 10 bit Analogue to Digital convertor (ADC) which has a reference voltage of 5 V.
The smallest current 𝐼𝑑 which can be measured by the circuit in nano-Amperes (rounded
off to one decimal place) is _____

Q.61 A capacitor is made of two circular metal plates of radius 1 m separated by a distance of
𝑑 = 1 mm. The space between them is filled by a dielectric with permittivity 𝜖𝑟 = 5. The
capacitor is connected to a voltage 𝑉 = 10 sin(2𝜋 × 106 × 𝑡) volts, where 𝑡 is in seconds.
The maximum value of the magnetic field in between the plates at a radial distance
𝑟 = 0.5 m from the centre of the capacitor is 𝐵 × 10−6 T. The value of 𝐵 (rounded off to
two decimal places) is _____

(Speed of light in vacuum 𝑐 = 3 × 108 m. s −1 )

Q.62 Rotational spectrum of a diatomic molecule consists of lines of equal spacing with an
interval of 20.0 cm−1 . Its moment of inertia is found to be 𝐼0 × 10−47 kg. m2, the value of
𝐼0 (rounded off to one decimal place) is _____

(ℎ = 6.6 × 10−34 J. s, speed of light in vacuum c = 3 × 108 m. s −1 )

Q.63 Copper has an electron number density of 8.3 × 1028 m−3 . Its Fermi energy in eV
(rounded off to one decimal place) is _____

(ℏ = 1.06 × 10−34 J. s, mass of electron me = 9.10 × 10−31 kg, charge of electron =
1.60 × 10−19 C)

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Q.64
Two 1 kg blocks are connected to two massless springs of spring constants 8 N. m−1 and
4 N. m−1 . The system is kept on a frictionless horizontal floor with one end of a spring
attached to a wall (see figure below). They are performing oscillatory motion along the
𝜔
𝑥-axis with the normal mode frequencies 𝜔𝐻 and 𝜔𝐿 (𝜔𝐻 > 𝜔𝐿 ). The ratio 𝜔𝐻 (rounded
𝐿
off to two decimal places) is _____
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Q.65 A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table
and the other end resting on one’s index finger. As the finger is removed (see figure below),
the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated
by a negligibly small angle but has gained a rotational speed as it leaves the table and falls
vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated
by an angle 𝜃. The value of 𝜃 in degrees (rounded off to one decimal place) is _____
(g = 9.8 m.s−2)

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