GATE 2021 Question Paper PH Physics

General Aptitude (GA)

Q.1 – Q.5 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 (i) Arun and Aparna are here.
(ii) Arun and Aparna is here.
(iii) Arun’s families is here.
(iv) Arun’s family is here.
Which of the above sentences are grammatically CORRECT?

(A) (i) and (ii)

(B) (i) and (iv)

(C) (ii) and (iv)

(D) (iii) and (iv)

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Q.2

The mirror image of the above text about the x-axis is

(A)

(B)

(C)

(D)

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Q.3 Two identical cube shaped dice each with faces numbered 1 to 6 are rolled
simultaneously. The probability that an even number is rolled out on each
dice is:

(A) 1
36
(B) 1
12
(C) 1
8
(D) 1
4

Q.4  and ⊙ are two operators on numbers p and q such that
𝒑 ⊙ 𝒒 = 𝒑 − 𝒒, and p  q = p  q
Then, (𝟗 ⊙ (𝟔 ⊕ 𝟕)) ⊙ (𝟕 ⊕ (𝟔 ⊙ 𝟓))=

(A) 40

(B) -26

(C) -33

(D) -40

Q.5 Four persons P, Q, R and S are to be seated in a row. R should not be seated
at the second position from the left end of the row. The number of distinct
seating arrangements possible is:

(A) 6

(B) 9

(C) 18

(D) 24

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Q. 6 – Q. 10 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong
answer: – 2/3).

Q.6 On a planar field, you travelled 3 units East from a point O. Next you
travelled 4 units South to arrive at point P. Then you travelled from P in the
North-East direction such that you arrive at a point that is 6 units East of
point O. Next, you travelled in the North-West direction, so that you arrive
at point Q that is 8 units North of point P.
The distance of point Q to point O, in the same units, should be _____

(A) 3

(B) 4

(C) 5

(D) 6

Q.7 The author said, “Musicians rehearse before their concerts. Actors rehearse
their roles before the opening of a new play. On the other hand, I find it
strange that many public speakers think they can just walk on to the stage
and start speaking. In my opinion, it is no less important for public speakers
to rehearse their talks.”
Based on the above passage, which one of the following is TRUE?

(A) The author is of the opinion that rehearsing is important for musicians, actors
and public speakers.

(B) The author is of the opinion that rehearsing is less important for public speakers
than for musicians and actors.

(C) The author is of the opinion that rehearsing is more important only for
musicians than public speakers.

(D) The author is of the opinion that rehearsal is more important for actors than
musicians.

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Q.8 1. Some football players play cricket.
2. All cricket players play hockey.
Among the options given below, the statement that logically follows from
the two statements 1 and 2 above, is:

(A) No football player plays hockey.

(B) Some football players play hockey.

(C) All football players play hockey.

(D) All hockey players play football.

Q.9

In the figure shown above, PQRS is a square. The shaded portion is formed
by the intersection of sectors of circles with radius equal to the side of the
square and centers at S and Q.
The probability that any point picked randomly within the square falls in the
shaded area is ______

(A) 4 − 𝜋
2
(B) 1
2
(C) 𝜋 − 1
2
(D) 𝜋
4

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Q.10 In an equilateral triangle PQR, side PQ is divided into four equal parts, side
QR is divided into six equal parts and side PR is divided into eight equal parts.
The length of each subdivided part in cm is an integer.
The minimum area of the triangle PQR possible, in cm2, is

(A) 18

(B) 24

(C) 48√3

(D) 144√3

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Q.1 – Q.9 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 Choose the graph that best describes the variation of dielectric constant (𝝐𝒓 )
with temperature (𝑻) in a ferroelectric material.
(𝑻𝑪 is the Curie temperature)

(A)

(B)

(C)

(D)

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Q.2 A matter wave is represented by the wave function
𝚿(𝒙, 𝒚, 𝒛, 𝒕) = 𝑨𝒆𝒊 (𝟒𝒙+𝟑𝒚+𝟓𝒛−𝟏𝟎𝝅𝒕)
where 𝑨 is a constant. The unit vector representing the direction of
propagation of this matter wave is

(A) 4 3 1
𝑥̂ + 𝑦̂ + 𝑧̂
5√2 5√2 √2

(B) 3 4 1
𝑥̂ + 𝑦̂ + 𝑧̂
5√2 5√2 5√2

(C) 1 3 1
𝑥̂ + 𝑦̂ + 𝑧̂
5√2 5√2 √2

(D) 1 4 3
𝑥̂ + 𝑦̂ + 𝑧̂
5√2 5√2 5√2

Q.3 As shown in the figure, X-ray diffraction pattern is obtained from a diatomic
chain of atoms 𝑷 and 𝑸. The diffraction condition is given by 𝒂 𝐜𝐨𝐬 𝜽 = 𝒏𝝀,
where 𝒏 is the order of the diffraction peak. Here, 𝒂 is the lattice constant
and 𝝀 is the wavelength of the X-rays. Assume that atomic form factors and
resolution of the instrument do not depend on 𝜽. Then, the intensity of the
diffraction peaks is

(A) lower for even values of 𝑛, when compared to odd values of 𝑛

(B) lower for odd values of 𝑛, when compared to even values of 𝑛

(C) zero for odd values of 𝑛

(D) zero for even values of 𝑛

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Q.4 As shown in the figure, two metal-semiconductor junctions are formed
between an n-type semiconductor 𝑺 and metal 𝑴. The work functions of 𝑺
and 𝑴 are 𝝋𝑺 𝐚𝐧𝐝 𝝋𝑴 , respectively with 𝝋𝑴 > 𝝋𝑺 .

The 𝑰 − 𝑽 characteristics (on linear scale) of the junctions is best
represented by

(A)

(B)

(C)

(D)

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Q.5 Consider a tiny current loop driven by a sinusoidal alternating current. If
the surface integral of its time-averaged Poynting vector is constant, then the
magnitude of the time-averaged magnetic field intensity, at any arbitrary
⃗ , is proportional to
position, 𝒓

(A) 1
𝑟3
(B) 1
𝑟2
(C) 1
𝑟
(D) 𝑟

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Q.6 Consider a solenoid of length 𝑳 and radius 𝑹, where 𝑹 ≪ 𝑳. A steady-current
flows through the solenoid. The magnetic field is uniform inside the solenoid
and zero outside.

Among the given options, choose the one that best represents the variation
in the magnitude of the vector potential, (𝟎, 𝑨𝝋 , 𝟎) at 𝒛 = 𝑳/𝟐, as a function
of the radial distance (𝒓) in cylindrical coordinates.
Useful information: The curl of a vector ⃗𝑭, in cylindrical coordinates is
𝟏 𝝏(𝒓𝑭𝝋 )
⃗ (𝒓, 𝝋, 𝒛) = 𝒓̂ [𝟏 𝝏𝑭𝒛 − 𝝏𝑭𝝋 ] + 𝝋
⃗ ×𝑭
𝛁
𝝏𝑭 𝝏𝑭
̂ [ 𝒓 − 𝒛 ] + 𝒛̂ [ −
𝝏𝑭𝒓
]
𝒓 𝝏𝝋 𝝏𝒛 𝝏𝒛 𝝏𝒓 𝒓 𝝏𝒓 𝝏𝝋

(A)

(B)

(C)

(D)

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Q.7 Assume that 13N (𝒁 = 𝟕) undergoes first forbidden 𝜷+ decay from its ground
state with spin-parity 𝑱𝝅𝒊 , to a final state 𝑱𝝅𝒇 . The possible values for 𝑱𝝅𝒊 and
𝑱𝝅𝒇 , respectively, are

(A) 1− 5+
,
2 2
(B) 1+ 5+
,
2 2
− −
(C) 1 1
,
2 2
(D) 1+ 1−
,
2 2

Q.8 In an experiment, it is seen that an electric-dipole (𝑬𝟏) transition can
connect an initial nuclear state of spin-parity 𝑱𝝅𝒊 = 𝟐+ to a final state 𝑱𝝅𝒇 . All
possible values of 𝑱𝝅𝒇 are

(A) 1+ , 2+

(B) 1+ , 2+ , 3+

(C) 1− , 2−

(D) 1− , 2− , 3−

Q.9 Choose the correct statement from the following.

(A) Silicon is a direct band gap semiconductor.

(B) Conductivity of metals decreases with increase in temperature.

(C) Conductivity of semiconductors decreases with increase in temperature.

(D) Gallium Arsenide is an indirect band gap semiconductor.

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Q.10 – Q.16 Multiple Select Question (MSQ), carry ONE mark each (no negative marks).

Q.10 A two-dimensional square lattice has lattice constant 𝒂.
𝒌 represents the wavevector in reciprocal space. The coordinates (𝒌𝒙 , 𝒌𝒚 )
of reciprocal space where band gap(s) can occur, are

(A) (0, 0)

(B) (± 𝜋 , ± 𝜋)
𝑎 𝑎
(C) (± 𝜋 , ± 𝜋 )
𝑎 1.3𝑎
(D) (± 𝜋 , ± 𝜋)
3𝑎 𝑎

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Q.11 As shown in the figure, an electromagnetic wave with intensity 𝑰𝑰 is incident
at the interface of two media having refractive indices 𝒏𝟏 = 𝟏 and 𝒏𝟐 = √𝟑.
The wave is reflected with intensity 𝑰𝑹 and transmitted with intensity 𝑰𝑻 .
Permeability of each medium is the same. (Reflection coefficient 𝑹 = 𝑰𝑹 /𝑰𝑰
and Transmission coefficient 𝑻 = 𝑰𝑻 /𝑰𝑰 ).

Choose the correct statement(s).

(A) 𝑅 = 0 if 𝜃𝐼 = 0o and polarization of incident light is parallel to the plane of
incidence.

(B) 𝑇 = 1 if 𝜃𝐼 = 60o and polarization of incident light is parallel to the plane of
incidence.

(C) 𝑅 = 0 if 𝜃𝐼 = 60o and polarization of incident light is perpendicular to the
plane of incidence.

(D) 𝑇 = 1 if 𝜃𝐼 = 60o and polarization of incident light is perpendicular to the
plane of incidence.

Q.12 A material is placed in a magnetic field intensity 𝑯. As a result, bound
current density 𝑱𝒃 is induced and magnetization of the material is 𝑴. The
magnetic flux density is 𝑩. Choose the correct option(s) valid at the surface
of the material.

(A) 𝛁. 𝑴 = 0

(B) 𝛁. 𝑩 = 0

(C) 𝛁. 𝑯 = 0

(D) 𝛁. 𝑱𝑏 = 0

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Q.13 For a finite system of Fermions where the density of states increases with
energy, the chemical potential

(A) decreases with temperature

(B) increases with temperature

(C) does not vary with temperature

(D) corresponds to the energy where the occupation probability is 0.5

Q.14 Among the term symbols
𝟒 𝟐 𝟑
𝑺𝟏 , 𝑫𝟕/𝟐 , 𝑺𝟏 and 𝟐𝑫𝟓/𝟐
choose the option(s) possible in the 𝑳𝑺 coupling notation.
4
(A) 𝑆1

(B) 2
𝐷7/2

(C) 3
𝑆1

(D) 2
𝐷5/2

Q.15 To sustain lasing action in a three-level laser as shown in the figure,
necessary condition(s) is(are)

(A) lifetime of the energy level 1 should be greater than that of energy level 2

(B) population of the particles in level 1 should be greater than that of level 0

(C) lifetime of the energy level 2 should be greater than that of energy level 0

(D) population of the particles in level 2 should be greater than that of level 1

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Q.16 If 𝒚𝒏 (𝒙) is a solution of the differential equation
𝒚′′ − 𝟐𝒙𝒚′ + 𝟐𝒏𝒚 = 𝟎
where 𝒏 is an integer and the prime ( ′ ) denotes differentiation with
𝟐
respect to 𝒙, then acceptable plot(s) of 𝝍𝒏 (𝒙) = 𝒆−𝒙 ⁄𝟐 𝒚𝒏 (𝒙), is(are)

(A)

(B)

(C)

(D)

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Q.17 – Q.25 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.17 The donor concentration in a sample of n-type silicon is increased by a factor
of 100. Assuming the sample to be non-degenerate, the shift in the Fermi
level (in meV) at 300 K (rounded off to the nearest integer) is ______.
(Given: 𝒌𝐁 𝑻= 25 meV at 300 K)

Q.18 Two observers O and Oʹ observe two events P and Q. The observers have a
constant relative speed of 0.5c. In the units, where the speed of light, c, is
taken as unity, the observer O obtained the following coordinates:
Event P: 𝒙 = 5, 𝒚 = 3, 𝒛 = 5, 𝒕=3
Event Q: 𝒙 = 5, 𝒚 = 1, 𝒛 = 3, 𝒕=5
The length of the space-time interval between these two events, as
measured by Oʹ, is 𝑳. The value of |𝑳| (in integer) is______.

Q.19 A light source having its intensity peak at the wavelength 289.8 nm is
calibrated as 10,000 K which is the temperature of an equivalent black body
radiation. Considering the same calibration, the temperature of light source
(in K) having its intensity peak at the wavelength 579.6 nm (rounded off to
the nearest integer) is ______.

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Q.20 A hoop of mass 𝑴 and radius 𝑹 rolls without slipping along a straight line
on a horizontal surface as shown in the figure. A point mass 𝒎 slides without
friction along the inner surface of the hoop, performing small oscillations
about the mean position. The number of degrees of freedom of the system
(in integer) is ______.

Q.21 Three non-interacting bosonic particles of mass 𝒎 each, are in a one-
dimensional infinite potential well of width 𝒂. The energy of the third excited
ħ𝟐 𝝅𝟐
state of the system is 𝒙 × 𝒎𝒂𝟐 . The value of 𝒙 (in integer) is ______.

Q.22 The spacing between two consecutive S-branch lines of the rotational Raman
spectra of hydrogen gas is 𝟐𝟒𝟑. 𝟐 𝐜𝐦−𝟏 . After excitation with a laser of
wavelength 𝟓𝟏𝟒. 𝟓 𝐧𝐦, the Stoke’s line appeared at 𝟏𝟕𝟔𝟏𝟏. 𝟒 𝐜𝐦−𝟏 for a
particular energy level. The wavenumber (rounded off to the nearest
integer), in 𝐜𝐦−𝟏 , at which Stoke’s line will appear for the next higher energy
level is ______.

Q.23 The transition line, as shown in the figure, arises between
𝟐
𝑫𝟑/𝟐 and 𝟐𝑷𝟏/𝟐 states without any external magnetic field. The number of
lines that will appear in the presence of a weak magnetic field (in integer) is
______.

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Q.24 Consider the atomic system as shown in the figure, where the Einstein 𝑨
coefficients for spontaneous emission for the levels are 𝑨𝟐→𝟏 = 𝟐 × 𝟏𝟎𝟕 s–1
and 𝑨𝟏→𝟎 = 𝟏𝟎𝟖 s–1. If 1014 atoms/cm3 are excited from level 0 to level 2 and
a steady state population in level 2 is achieved, then the steady state
population at level 1 will be 𝒙 × 𝟏𝟎𝟏𝟑 cm–3. The value of 𝒙 (in integer) is
______.

Q.25 If 𝒂 ⃗ are constant vectors, 𝒓
⃗ and 𝒃 ⃗ and 𝒑 ⃗ are generalized positions and
conjugate momenta, respectively, then for the transformation 𝑸 = 𝒂 ⃗ ∙𝒑
⃗ and
⃗
𝑷= 𝒃∙𝒓 ⃗ to be canonical, the value of 𝒂 ⃗
⃗ ∙ 𝒃 (in integer) is ______.

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Q.26 – Q.41 Multiple Choice Question (MCQ), carry TWO mark each (for each wrong
answer: – 2/3).

Q.26

The above combination of logic gates represents the operation

(A) OR

(B) NAND

(C) AND

(D) NOR

Q.27 In a semiconductor, the ratio of the effective mass of hole to electron is 2:11
and the ratio of average relaxation time for hole to electron is 1:2. The ratio
of the mobility of the hole to electron is

(A) 4:9

(B) 4:11

(C) 9:4

(D) 11:4

Q.28 𝟏 𝟐+𝒊
Consider a spin 𝑺 = ℏ⁄𝟐 particle in the state |𝝓⟩ = 𝟑 [ ]. The probability
𝟐
that a measurement finds the state with 𝑺𝒙 = +ℏ⁄𝟐 is

(A) 5/18

(B) 11/18

(C) 15/18

(D) 17/18

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Q.29 An electromagnetic wave having electric field 𝑬 = 𝟖 𝐜𝐨𝐬(𝒌𝒛 − 𝝎𝒕) 𝒚 ̂ 𝐕 𝐜𝐦−𝟏
is incident at 𝟗𝟎° (normal incidence) on a square slab from vacuum (with
refractive index 𝒏𝟎 = 𝟏. 𝟎) as shown in the figure. The slab is composed of
two different materials with refractive indices 𝒏𝟏 and 𝒏𝟐 . Assume that the
permeability of each medium is the same. After passing through the slab for
the first time, the electric field amplitude, in 𝐕 𝐜𝐦−𝟏 , of the electromagnetic
wave, which emerges from the slab in region 2, is closest to

(A) 11
1.6
(B) 11
3.2
(C) 11
13.8
(D) 11
25.6

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Q.30 Consider a point charge +𝑸 of mass 𝒎 suspended by a massless, inextensible
string of length 𝒍 in free space (permittivity 𝜺𝟎 ) as shown in the figure. It is
placed at a height 𝒅 (𝒅 > 𝒍) over an infinitely large, grounded conducting
plane. The gravitational potential energy is assumed to be zero at the
position of the conducting plane and is positive above the plane.

If 𝜽 represents the angular position and 𝒑𝜽 its corresponding canonical
momentum, then the correct Hamiltonian of the system is

(A) 𝑝𝜃2 𝑄2
− − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
2𝑚𝑙 2 16𝜋𝜀0 (𝑑 − 𝑙 cos 𝜃)

(B) 𝑝𝜃2 𝑄2
− + 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
2𝑚𝑙 2 8𝜋𝜀0 (𝑑 − 𝑙 cos 𝜃)

(C) 𝑝𝜃2 𝑄2
− − 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
2𝑚𝑙 2 8𝜋𝜀0 (𝑑 − 𝑙 cos 𝜃)

(D) 𝑝𝜃2 𝑄2
− + 𝑚𝑔(𝑑 − 𝑙 cos 𝜃)
2𝑚𝑙 2 16𝜋𝜀0 (𝑑 − 𝑙 cos 𝜃)

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Q.31 Consider two concentric conducting spherical shells as shown in the figure.
The inner shell has a radius a and carries a charge +𝑸. The outer shell has
a radius b and carries a charge −𝑸. The empty space between them is half-
filled by a hemispherical shell of a dielectric having permittivity 𝜺𝟏 . The
remaining space between the shells is filled with air having the permittivity
𝜺𝟎 .

The electric field at a radial distance 𝒓 from the center and between the shells
(𝒂 < 𝒓 < 𝒃) is

(A) 𝑄 𝑟̂
everywhere
2 𝜋 (𝜀0 +𝜀1 ) 𝑟 2

(B) 𝑄 𝑟̂ 𝑄 𝑟̂
on the air side and on the dielectric side
4 𝜋 𝜀0 𝑟 2 4 𝜋 𝜀1 𝑟 2

(C) 𝑄 𝑟̂ 𝑄 𝑟̂
on the air side and on the dielectric side
2 𝜋 𝜀0 𝑟2 2 𝜋 𝜀1 𝑟 2

(D) 𝑄 𝑟̂
everywhere
4 𝜋 (𝜀0 +𝜀1 ) 𝑟 2

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Q.32

For the given sets of energy levels of nuclei X and Y whose mass numbers
are odd and even, respectively, choose the best suited interpretation.

(A) Set I: Rotational band of X
Set II: Vibrational band of Y

(B) Set I: Rotational band of Y
Set II: Vibrational band of X

(C) Set I: Vibrational band of X
Set II: Rotational band of Y

(D) Set I: Vibrational band of Y
Set II: Rotational band of X

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Q.33 Consider a system of three distinguishable particles, each having spin 𝑺 =
𝟏/𝟐 such that 𝑺𝒛 = ±𝟏/𝟐 with corresponding magnetic moments 𝝁𝒛 = ±𝝁.
When the system is placed in an external magnetic field 𝑯 pointing along the
𝒛-axis, the total energy of the system is 𝝁𝑯. Let 𝒙 be the state where the first
spin has 𝑺𝒛 = 𝟏/𝟐. The probability of having the state 𝒙 and the mean
magnetic moment (in the +𝒛 direction) of the system in state 𝒙 are

(A) 1 −1
, 𝜇
3 3
(B) 1 2
, 𝜇
3 3
(C) 2 −2
, 𝜇
3 3
(D) 2 1
, 𝜇
3 3

Q.34 Consider a particle in a one-dimensional infinite potential well with its walls
at 𝒙 = 𝟎 and 𝒙 = 𝑳. The system is perturbed as shown in the figure.

The first order correction to the energy eigenvalue is

(A) 𝑉0
4
(B) 𝑉0
3
(C) 𝑉0
2
(D) 𝑉0
5

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Q.35 Consider a state described by 𝝍(𝒙, 𝒕) = 𝝍𝟐 (𝒙, 𝒕) + 𝝍𝟒 (𝒙, 𝒕), where 𝝍𝟐 (𝒙, 𝒕)
and 𝝍𝟒 (𝒙, 𝒕) are respectively the second and fourth normalized harmonic
oscillator wave functions and 𝝎 is the angular frequency of the harmonic
oscillator. The wave function 𝝍(𝒙, 𝒕 = 𝟎) will be orthogonal to 𝝍(𝒙, 𝒕) at
time 𝒕 equal to

(A) 𝜋
2𝜔
(B) 𝜋
𝜔
(C) 𝜋
4𝜔
(D) 𝜋
6𝜔

Q.36 Consider a single one-dimensional harmonic oscillator of angular frequency
𝝎, in equilibrium at temperature 𝑻 = (𝒌𝐁 𝜷)−𝟏. The states of the harmonic
𝟏
oscillator are all non-degenerate having energy 𝑬𝒏 = (𝒏 + 𝟐) ℏ𝝎 with equal
probability, where 𝒏 is the quantum number. The Helmholtz free energy of
the oscillator is

(A) ħ𝜔
+ 𝛽 −1 ln[1 − exp(𝛽ħ𝜔)]
2
(B) ħ𝜔
+ 𝛽 −1 ln[1 − exp(−𝛽ħ𝜔)]
2
(C) ħ𝜔
+ 𝛽 −1 ln[1 + exp(−𝛽ħ𝜔)]
2
(D) 𝛽 −1 ln[1 − exp(−𝛽ħ𝜔)]

PH - Copyright © GATE 2021 Page 26 of 37

Q.37 A system of two atoms can be in three quantum states having energies 0, 𝜖 and
2𝜖. The system is in equilibrium at temperature 𝑇 = (𝑘𝐵 𝛽)−1 . Match the
following Statistics with the Partition function.

Statistics Partition function

CD: Classical 𝐙𝟏: 𝑒 −𝛽𝜖 + 𝑒 −2𝛽𝜖 + 𝑒 −3𝛽𝜖
(distinguishable particles)

CI: Classical 𝐙𝟐: 1 + 𝑒 −𝛽𝜖 + 2𝑒 −2𝛽𝜖 + 𝑒 −3𝛽𝜖 + 𝑒 −4𝛽𝜖
(indistinguishable particles)

FD: Fermi-Dirac 𝐙𝟑: 1 + 2𝑒 −𝛽𝜖 + 3𝑒 −2𝛽𝜖 + 2𝑒 −3𝛽𝜖 + 𝑒 −4𝛽𝜖

BE: Bose-Einstein 1 3 1
𝐙𝟒: + 𝑒 −𝛽𝜖 + 𝑒 −2𝛽𝜖 + 𝑒 −3𝛽𝜖 + 𝑒 −4𝛽𝜖
2 2 2

(A) CD:Z1, CI:Z2, FD:Z3, BE:Z4

(B) CD:Z2, CI:Z3, FD:Z4, BE:Z1

(C) CD:Z3, CI:Z4, FD:Z1, BE:Z2

(D) CD:Z4, CI:Z1, FD:Z2, BE:Z3

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Q.38 The free energy of a ferromagnet is given by 𝑭 = 𝑭𝟎 + 𝒂𝟎 (𝑻 − 𝑻𝑪 )𝑴𝟐 +
𝒃𝑴𝟒 , where 𝑭𝟎 , 𝒂𝟎 , and 𝒃 are positive constants, 𝑴 is the magnetization, 𝑻
is the temperature, and 𝑻𝑪 is the Curie temperature. The relation between
𝑴𝟐 and 𝑻 is best depicted by

(A)

(B)

(C)

(D)

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Q.39 Consider a spherical galaxy of total mass 𝑴 and radius 𝑹, having a uniform
matter distribution. In this idealized situation, the orbital speed 𝒗 of a star
of mass 𝒎 (𝒎 ≪ 𝑴) as a function of the distance 𝒓 from the galactic centre
is best described by
(𝑮 is the universal gravitational constant)

(A)

(B)

(C)

(D)

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Q.40 Consider the potential 𝑼(𝒓) defined as
𝒆−𝜶𝒓
𝑼(𝒓) = −𝑼𝟎
𝒓
where 𝜶 and 𝑼𝟎 are real constants of appropriate dimensions. According to
the first Born approximation, the elastic scattering amplitude calculated
with 𝑼(𝒓) for a (wave-vector) momentum transfer 𝒒 and 𝜶 → 𝟎, is
proportional to
∞ 𝒒
(Useful integral: ∫𝟎 𝐬𝐢𝐧(𝒒𝒓)𝒆−𝜶𝒓 𝒅𝒓 = 𝜶𝟐+𝒒𝟐 )

(A) 𝑞 −2

(B) 𝑞 −1

(C) 𝑞

(D) 𝑞 2

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Q.41 As shown in the figure, inverse magnetic susceptibility (𝟏/𝝌) is plotted as a
function of temperature (𝑻) for three different materials in paramagnetic
states.

(Curie temperature of ferromagnetic material =𝑻𝐂
Néel temperature of antiferromagnetic material= 𝑻𝐍 )
Choose the correct statement from the following

(A) Material 1 is antiferromagnetic (𝑇 < 𝑇N ), 2 is paramagnetic, and 3 is
ferromagnetic (𝑇 < 𝑇C ).

(B) Material 1 is paramagnetic, 2 is antiferromagnetic (𝑇 < 𝑇N ), and 3 is
ferromagnetic (𝑇 < 𝑇C ).

(C) Material 1 ferromagnetic (𝑇 < 𝑇C ), 2 is antiferromagnetic (𝑇 < 𝑇N ), and 3 is
paramagnetic.

(D) Material 1 is ferromagnetic (𝑇 < 𝑇C ), 2 is paramagnetic, and 3 is
antiferromagnetic (𝑇 < 𝑇N ).

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Q.42 – Q.46 Multiple Select Question (MSQ), carry TWO mark each (no negative marks).

Q.42 A function 𝒇(𝒕) is defined only for 𝒕 ≥ 𝟎. The Laplace transform of 𝒇(𝒕) is
∞
𝓛(𝒇; 𝒔) = ∫ 𝒆−𝒔𝒕 𝒇(𝒕) 𝒅𝒕
𝟎
whereas the Fourier transform of 𝒇(𝒕) is
∞
𝒇̃(𝝎) = ∫ 𝒇(𝒕) 𝒆−𝒊𝝎𝒕 𝒅𝒕 .
𝟎
The correct statement(s) is(are)

(A) The variable 𝑠 is always real.

(B) The variable 𝑠 can be complex.

(C) ℒ(𝑓; 𝑠) and 𝑓̃(𝜔) can never be made connected.

(D) ℒ(𝑓; 𝑠) and 𝑓̃(𝜔) can be made connected.

Q.43 𝑷 and 𝑸 are two Hermitian matrices and there exists a matrix 𝑹, which
diagonalizes both of them, such that 𝑹𝑷𝑹−𝟏 = 𝑺𝟏 and 𝑹𝑸𝑹−𝟏 = 𝑺𝟐 , where
𝑺𝟏 and 𝑺𝟐 are diagonal matrices. The correct statement(s) is(are)

(A) All the elements of both matrices 𝑆1 and 𝑆2 are real.

(B) The matrix 𝑃𝑄 can have complex eigenvalues.

(C) The matrix 𝑄𝑃 can have complex eigenvalues.

(D) The matrices 𝑃 and 𝑄 commute.

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Q.44 A uniform block of mass 𝑴 slides on a smooth horizontal bar. Another mass
𝒎 is connected to it by an inextensible string of length l of negligible mass,
and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the
masses. The number of degrees of freedom of the system is two and the
generalized coordinates are chosen as x and , as shown in the figure.

If 𝒑𝒙 and 𝒑𝜽 are the generalized momenta corresponding to x and ,
respectively, then the correct option(s) is(are)

(A) 𝑝𝑥 = (𝑚 + 𝑀)𝑥̇ + 𝑚𝑙 cos 𝜃 𝜃̇

(B) 𝑝𝜃 = 𝑚𝑙 2 𝜃̇ − 𝑚𝑙 cos 𝜃 𝑥̇

(C) 𝑝𝑥 is conserved

(D) 𝑝𝜃 is conserved

Q.45 The Gell-Mann – Okuba mass formula defines the mass of baryons as
𝟏
𝑴 = 𝑴𝟎 + 𝒂𝒀 + 𝒃 [𝑰(𝑰 + 𝟏) − 𝟒 𝒀𝟐 ], where 𝑴𝟎 , 𝒂 and 𝒃 are constants,
𝑰 represents the isospin and 𝒀 represents the hypercharge. If the mass of
𝚺 hyperons is same as that of 𝚲 hyperons, then the correct option(s) is(are)

(A) 𝑀 ∝ 𝐼(𝐼 + 1)

(B) 𝑀 ∝ 𝑌

(C) 𝑀 does not depend on 𝐼

(D) 𝑀 does not depend on 𝑌

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Q.46 The time derivative of a differentiable function 𝒈(𝒒𝒊 , 𝒕) is added to a
Lagrangian 𝑳(𝒒𝒊 , 𝒒̇𝒊 , 𝒕) such that
𝒅
𝑳′ = 𝑳(𝒒𝒊 , 𝒒̇ 𝒊 , 𝒕) +
𝒈(𝒒𝒊 , 𝒕)
𝒅𝒕
where 𝒒𝒊 , 𝒒̇ 𝒊 , 𝒕 are the generalized coordinates, generalized velocities and
time, respectively. Let 𝒑𝒊 be the generalized momentum and 𝑯 the
Hamiltonian associated with 𝑳(𝒒𝒊 , 𝒒̇ 𝒊 , 𝒕). If 𝒑′𝒊 and 𝑯′ are those associated
with 𝑳′ , then the correct option(s) is(are)

(A) Both 𝐿 and 𝐿′ satisfy the Euler-Lagrange’s equations of motion

(B) ∂
𝑝𝑖′ = 𝑝𝑖 + 𝑔(𝑞𝑖 , 𝑡)
∂𝑞𝑖

(C) If 𝑝𝑖 is conserved, then 𝑝𝑖′ is necessarily conserved

(D) 𝑑
𝐻′ = 𝐻 + 𝑔(𝑞𝑖 , 𝑡)
𝑑𝑡

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Q.47 – Q.55 Numerical Answer Type (NAT), carry TWO mark each (no negative
marks).

Q.47 A linear charged particle accelerator is driven by an alternating voltage
source operating at 10 MHz. Assume that it is used to accelerate electrons.
After a few drift-tubes, the electrons attain a velocity 𝟐. 𝟗 × 𝟏𝟎𝟖 m s–1. The
minimum length of each drift-tube, in m, to accelerate the electrons further
(rounded off to one decimal place) is______.

Q.48 The Coulomb energy component in the binding energy of a nucleus is
18.432 MeV. If the radius of the uniform and spherical charge distribution
in the nucleus is 3 fm, the corresponding atomic number (rounded off to the
nearest integer) is _______.
𝒆𝟐
(Given: 𝟒𝝅𝝐 = 𝟏. 𝟒𝟒 MeV fm)
𝟎

Q.49 For a two-nucleon system in spin singlet state, the spin is represented
through the Pauli matrices 𝝈𝟏 , 𝝈𝟐 for particles 1 and 2, respectively.
The value of (𝝈𝟏 ⋅ 𝝈𝟐 ) (in integer) is ________.

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Q.50 A contour integral is defined as
𝒅𝒛
𝑰𝒏 = ∮
𝑪 (𝒛 − 𝒏)𝟐 + 𝝅𝟐
where 𝒏 is a positive integer and 𝑪 is the closed contour, as shown in the
figure, consisting of the line from −𝟏𝟎𝟎 to 𝟏𝟎𝟎 and the semicircle traversed
in the counter-clockwise sense.

The value of ∑𝟓𝒏=𝟏 𝑰𝒏 (in integer) is ______.

Q.51 The normalized radial wave function of the second excited state of hydrogen
atom is
𝟏 −𝟑⁄𝒓 −𝒓⁄
𝑹(𝒓) = 𝒂 𝟐 𝒆 𝟐𝒂
√𝟐𝟒 𝒂
where 𝒂 is the Bohr radius and 𝒓 is the distance from the center of the atom.
The distance at which the electron is most likely to be found is 𝒚 × 𝒂. The
value of 𝒚 (in integer) is ______.

Q.52 Consider an atomic gas with number density 𝒏 = 𝟏𝟎𝟐𝟎 𝐦−𝟑 , in the ground
state at 300 K. The valence electronic configuration of atoms is 𝒇𝟕 . The
paramagnetic susceptibility of the gas 𝝌 = 𝒎 × 𝟏𝟎−𝟏𝟏 . The value of 𝒎
(rounded off to two decimal places) is ______.
(Given: 𝐌𝐚𝐠𝐧𝐞𝐭𝐢𝐜 𝐩𝐞𝐫𝐦𝐞𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐟𝐫𝐞𝐞 𝐬𝐩𝐚𝐜𝐞 𝝁𝟎 = 𝟒𝝅 × 𝟏𝟎−𝟕 𝐇 𝐦−𝟏
𝐁𝐨𝐡𝐫 𝐦𝐚𝐠𝐧𝐞𝐭𝐨𝐧 𝝁𝐁 = 𝟗. 𝟐𝟕𝟒 × 𝟏𝟎−𝟐𝟒 𝐀 𝐦𝟐
𝐁𝐨𝐥𝐭𝐳𝐦𝐚𝐧𝐧 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝒌𝐁 = 𝟏. 𝟑𝟖𝟎𝟕 × 𝟏𝟎−𝟐𝟑 𝐉 𝐊 −𝟏 )

PH - Copyright © GATE 2021 Page 36 of 37

Q.53 Consider a cross-section of an electromagnet having an air-gap of 𝟓 cm as
shown in the figure. It consists of a magnetic material (𝝁 = 𝟐𝟎𝟎𝟎𝟎𝝁𝟎 ) and is
driven by a coil having 𝑵𝑰 = 𝟏𝟎𝟒 A, where 𝑵 is the number of turns and 𝑰 is
the current in Ampere.

⃗⃗ (in Tesla,
Ignoring the fringe fields, the magnitude of the magnetic field 𝑩
rounded off to two decimal places) in the air-gap between the magnetic poles
is ______.

Q.54 The spin 𝑺⃗ and orbital angular momentum 𝑳 ⃗ of an atom precess about 𝑱, the
total angular momentum. 𝑱 precesses about an axis fixed by a magnetic field
⃗⃗⃗⃗⃗
𝑩𝟏 = 𝟐𝑩𝟎 𝒛̂ , where 𝑩𝟎 is a constant. Now the magnetic field is changed to
⃗⃗⃗⃗⃗
𝑩𝟐 = 𝑩𝟎 (𝒙 ̂ + √𝟐𝒚
̂ + 𝒛̂). Given the orbital angular momentum quantum
number 𝒍 = 𝟐 and spin quantum number 𝒔 = 𝟏⁄𝟐, 𝜽 is the angle between
⃗⃗⃗⃗⃗
𝑩𝟏 and 𝑱 for the largest possible values of total angular quantum number 𝒋
and its z-component 𝒋𝒛 . The value of 𝜽 (in degree, rounded off to the nearest
integer) is______.

Q.55 The spin-orbit effect splits the 𝟐𝑷 → 𝟐𝑺 transition (wavelength, 𝝀 =
𝟔𝟓𝟐𝟏 Å) in Lithium into two lines with separation of ∆𝝀 = 𝟎. 𝟏𝟒 Å. The
corresponding positive value of energy difference between the above two
lines, in eV, is 𝒎 × 𝟏𝟎−𝟓. The value of 𝒎 (rounded off to the nearest
integer) is ______.
(Given: Planck’s constant, 𝐡 = 𝟒. 𝟏𝟐𝟓 × 𝟏𝟎−𝟏𝟓 𝐞𝐕 𝐬
Speed of light, 𝐜 = 𝟑 × 𝟏𝟎𝟖 𝐦 𝐬 −𝟏 )

END OF THE QUESTION PAPER

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