JAM 2022 Question Paper Physics (PH)

JAM 2022 PHYSICS - PH

Section A: Q.1 – Q.10 Carry ONE mark each.

Q.1 The equation 𝑧 2 + 𝑧̅ 2 = 4 in the complex plane (where 𝑧̅ is the complex
conjugate of 𝑧) represents

(A) Ellipse

(B) Hyperbola

(C) Circle of radius 2

(D) Circle of radius 4

Q.2 𝑐
A rocket (𝑆′) moves at a speed m/s along the positive x-axis, where c is the
2
speed of light. When it crosses the origin, the clocks attached to the rocket and
the one with a stationary observer (𝑆) located at 𝑥 = 0 are both set to zero. If 𝑆
observes an event at (𝑥, 𝑡), the same event occurs in the 𝑆′ frame at

(A) 𝑥 ′ = 2 (𝑥 − 𝑐𝑡) and 𝑡 ′ = 2 (𝑡 − 𝑥 )
√3 2 √3 2𝑐

(B) 𝑥 ′ = 2 (𝑥 + 𝑐𝑡) and 𝑡 ′ = 2 (𝑡 − 𝑥 )
√3 2 √3 2𝑐

(C) 𝑥 ′ = 2 (𝑥 − 𝑐𝑡) and 𝑡 ′ = 2 (𝑡 + 𝑥 )
√3 2 √3 2𝑐

(D) 𝑥 ′ = 2 (𝑥 + 𝑐𝑡) and 𝑡 ′ = 2 (𝑡 + 𝑥 )
√3 2 √3 2𝑐

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Q.3 Consider a classical ideal gas of 𝑁 molecules in equilibrium at temperature 𝑇.
Each molecule has two energy levels, −𝜖 and 𝜖. The mean energy of the gas is

(A) 0

(B) 𝑁𝜖 tanh ( 𝜖 )
𝑘𝐵 𝑇

(C) −𝑁𝜖 tanh ( 𝜖 )
𝑘𝐵 𝑇

(D) 𝜖
2

Q.4 At a temperature 𝑇, let 𝛽 and 𝜅 denote the volume expansivity and isothermal
𝛽
compressibility of a gas, respectively. Then is equal to
𝜅

(A) 𝜕𝑃
( )
𝜕𝑇 𝑉

(B) 𝜕𝑃
( )
𝜕𝑉 𝑇

(C) 𝜕𝑇
( )
𝜕𝑃 𝑉

(D) 𝜕𝑇
( )
𝜕𝑉 𝑃

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Q.5 The resultant of the binary subtraction 1110101 − 0011110 is

(A) 1001111

(B) 1010111

(C) 1010011

(D) 1010001

Q.6 Consider a particle trapped in a three-dimensional potential well such that
𝑈(𝑥, 𝑦, 𝑧) = 0 for 0 ≤ 𝑥 ≤ 𝑎, 0 ≤ 𝑦 ≤ 𝑎, 0 ≤ 𝑧 ≤ 𝑎 and 𝑈(𝑥, 𝑦, 𝑧) = ∞
everywhere else. The degeneracy of the 5th excited state is

(A) 1

(B) 3

(C) 6

(D) 9

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Q.7 A particle of mass 𝑚 and angular momentum 𝐿 moves in space where its
potential energy is

𝑈(𝑟) = 𝑘𝑟 2 (𝑘 > 0) and 𝑟 is the radial coordinate.

If the particle moves in a circular orbit, then the radius of the orbit is

(A) 1
𝐿2 4
( )
𝑚𝑘

(B) 1
𝐿2 4
( )
2𝑚𝑘

(C) 1
2
2𝐿 4
( )
𝑚𝑘

(D) 1
4𝐿2 4
( )
𝑚𝑘

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Q.8 Consider a two-dimensional force field

⃗⃗⃗ (𝑥, 𝑦) = (5𝑥 2 + 𝑎𝑦 2 + 𝑏𝑥𝑦) 𝑥̂ + (4𝑥 2 + 4𝑥𝑦 + 𝑦 2 ) 𝑦.
𝐹 ̂

If the force field is conservative, then the values of 𝑎 and 𝑏 are

(A) 𝑎 = 2 and 𝑏 = 4

(B) 𝑎 = 2 and 𝑏 = 8

(C) 𝑎 = 4 and 𝑏 = 2

(D) 𝑎 = 8 and 𝑏 = 2

Q.9 Consider an electrostatic field 𝐸⃗ in a region of space. Identify the INCORRECT
statement.

(A) The work done in moving a charge in a closed path inside the region is zero

(B) The curl of 𝐸⃗ is zero

(C) The field can be expressed as the gradient of a scalar potential

(D) The potential difference between any two points in the region is always zero

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Q.10 Which one of the following figures correctly depicts the intensity distribution for
Fraunhofer diffraction due to a single slit? Here, 𝒙 denotes the distance from the
centre of the central fringe and 𝑰 denotes the intensity.

(A)

(B)

(C)

(D)

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Section A: Q.11 – Q.30 Carry TWO marks each.

Q.11 The function 𝑓(𝑥) = 𝑒 sin 𝑥 is expanded as a Taylor series in 𝑥, around 𝑥 = 0,
in the form 𝑓(𝑥) = ∑∞ 𝑛
𝑛=0 𝑎𝑛 𝑥 . The value of 𝑎0 + 𝑎1 + 𝑎2 is

(A) 0

(B) 3
2

(C) 5
2

(D) 5

Q.12 Consider a unit circle 𝐶 in the 𝑥𝑦 plane, centered at the origin. The value of the
integral ∮[(sin 𝑥 − 𝑦)𝑑𝑥 − (sin 𝑦 − 𝑥)𝑑𝑦] over the circle 𝐶, traversed
anticlockwise, is

(A) 0

(B) 2𝜋

(C) 3𝜋

(D) 4𝜋

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Q.13 The current through a series 𝑅𝐿 circuit, subjected to a constant emf ℰ, obeys
𝑑𝑖
𝐿 𝑑𝑡 + 𝑖𝑅 = ℰ. Let 𝐿 = 1 𝑚𝐻, 𝑅 = 1 𝑘Ω and ℇ = 1 𝑉. The initial condition

is 𝑖(0) = 0. At 𝑡 = 1 µ𝑠, the current in mA is

(A) 1 − 2𝑒 −2

(B) 1 − 2𝑒 −1

(C) 1 − 𝑒 −1

(D) 2 − 2𝑒 −1

Q.14 An ideal gas in equilibrium at temperature 𝑇 expands isothermally to twice its
initial volume. If Δ𝑆, Δ𝑈 and Δ𝐹 denote the changes in its entropy, internal energy
and Helmholtz free energy respectively, then

(A) Δ𝑆 < 0, Δ𝑈 > 0, Δ𝐹 < 0

(B) Δ𝑆 > 0, Δ𝑈 = 0, Δ𝐹 < 0

(C) Δ𝑆 < 0, Δ𝑈 = 0, Δ𝐹 > 0

(D) Δ𝑆 > 0, Δ𝑈 > 0, Δ𝐹 = 0

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Q.15 In a dilute gas, the number of molecules with free path length ≥ 𝑥 is given by
𝑁(𝑥) = 𝑁0 𝑒 −𝑥/𝜆 , where 𝑁0 is the total number of molecules and 𝜆 is the mean
free path. The fraction of molecules with free path lengths between 𝜆 and 2𝜆 is

(A) 1
𝑒

(B) 𝑒
𝑒−1

(C) 𝑒2
𝑒−1

(D) 𝑒 − 1
𝑒2

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Q.16 Consider a quantum particle trapped in a one-dimensional potential well in the
region [−𝐿/2 < 𝑥 < 𝐿/2], with infinitely high barriers at 𝑥 = −𝐿/2 and
𝑥 = 𝐿/2. The stationary wave function for the ground state is 𝜓(𝑥) =
2 𝜋𝑥
√ cos ( ). The uncertainties in momentum and position satisfy
𝐿 𝐿

(A) Δ𝑝 = 𝜋ℏ and Δ𝑥 = 0
𝐿

(B) Δ𝑝 = 2𝜋ℏ and 0 < Δ𝑥 < 𝐿
𝐿 2√3

(C) Δ𝑝 = 𝜋ℏ and Δ𝑥 > 𝐿
𝐿 2√3

(D) Δ𝑝 = 0 and Δ𝑥 = 𝐿
2

Q.17 Consider a particle of mass 𝑚 moving in a plane with a constant radial speed 𝑟̇
and a constant angular speed 𝜃̇. The acceleration of the particle in (𝑟, 𝜃)
coordinates is

(A) 2𝑟𝜃̇ 2 𝑟̂ − 𝑟̇ 𝜃̇𝜃̂

(B) − 𝑟𝜃̇ 2 𝑟̂ + 2𝑟̇ 𝜃̇𝜃̂

(C) 𝑟̈ 𝑟̂ + 𝑟𝜃̈ 𝜃̂

(D) 𝑟̈ 𝜃𝑟̂ + 𝑟𝜃̈𝜃̂

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Q.18 A planet of mass 𝑚 moves in an elliptical orbit. Its maximum and minimum
distances from the Sun are 𝑅 and 𝑟, respectively. Let 𝐺 denote the universal
gravitational constant, and 𝑀 the mass of the Sun. Assuming 𝑀 >> 𝑚, the
angular momentum of the planet with respect to the center of the Sun is

(A)
2𝐺𝑀𝑅𝑟
𝑚√
(𝑅 + 𝑟)

(B)
𝐺𝑀𝑅𝑟
𝑚√
2(𝑅 + 𝑟)

(C)
𝐺𝑀𝑅𝑟
𝑚√
(𝑅 + 𝑟)

(D)
2𝐺𝑀𝑅𝑟
2𝑚√
(𝑅 + 𝑟)

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Q.19 Consider a conical region of height ℎ and base radius 𝑅 with its vertex at the
origin. Let the outward normal to its base be along the positive 𝑧-axis, as shown
⃗ = 𝐵0 𝑧̂ exists everywhere. Then the
in the figure. A uniform magnetic field, 𝐵
magnetic flux through the base (Φ𝑏 ) and that through the curved surface of the
cone (Φ𝑐 ) are

(A) Φ𝑏 = 𝐵0 𝜋𝑅 2 ; Φ𝑐 = 0

(B) 1 1
Φ𝑏 = − 2 𝐵0 𝜋𝑅 2 ; Φ𝑐 = 2 𝐵0 𝜋𝑅 2

(C) Φ𝑏 = 0 ; Φ𝑐 = − 𝐵0 𝜋𝑅 2

(D) Φ𝑏 = 𝐵0 𝜋𝑅 2 ; Φ𝑐 = −𝐵0 𝜋𝑅 2

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Q.20 Consider a thin annular sheet, lying on the 𝑥𝑦-plane, with 𝑅1 and 𝑅2 as its inner
and outer radii, respectively. If the sheet carries a uniform surface-charge density
𝜎 and spins about the origin 𝑂 with a constant angular velocity
𝜔
⃗ = 𝜔0 𝑧̂ then, the total current flow on the sheet is

(A) 2𝜋𝜎𝜔0 (𝑅2 3 − 𝑅1 3 )
3

(B) 𝜎𝜔0 (𝑅2 3 − 𝑅1 3 )

(C) 𝜋𝜎𝜔0 (𝑅2 3 − 𝑅1 3 )
3

(D) 2𝜋𝜎𝜔0 (𝑅2 − 𝑅1 )3
3

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Q.21 A radioactive nucleus has a decay constant 𝜆 and its radioactive daughter nucleus
has a decay constant 10𝜆. At time 𝑡 = 0, 𝑁0 is the number of parent nuclei and
there are no daughter nuclei present. 𝑁1 (𝑡) and 𝑁2 (𝑡) are the number of parent
and daughter nuclei present at time 𝑡, respectively.
The ratio 𝑁2 (𝑡)/𝑁1 (𝑡) is

(A) 1
[1 − 𝑒 −9𝜆𝑡 ]
9

(B) 1
[1 − 𝑒 −10𝜆𝑡 ]
10

(C) [1 − 𝑒 −10𝜆𝑡 ]

(D) [1 − 𝑒 −9𝜆𝑡 ]

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Q.22 ⃗ = 𝐵0 𝑧,
A uniform magnetic field 𝐵 ̂ where 𝐵0 > 0 exists as shown in the figure.
A charged particle of mass m and charge q (q > 0) is released at the origin, in the
⃗⃗⃗ directed at an angle 𝜃 = 45∘ with respect to the
yz-plane, with a velocity 𝑣
positive 𝑧-axis. Ignoring gravity, which one of the following is TRUE.

(A) The initial acceleration 𝑎 = 𝑞𝑣𝐵0 𝑥̂
𝑚
√2

(B) The initial acceleration 𝑎 = 𝑞𝑣𝐵0 𝑦̂
𝑚
√2

(C) The particle moves in a circular path

(D) The particle continues in a straight line with constant speed

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Q.23 For an ideal intrinsic semiconductor, the Fermi energy at 0 𝐾

(A) lies at the top of the valence band

(B) lies at the bottom of the conduction band

(C) lies at the center of the bandgap

(D) lies midway between center of the bandgap and bottom of the conduction band

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Q.24 A circular loop of wire with radius 𝑅 is centered at the origin of the 𝑥𝑦-plane.
⃗ (𝜌, 𝜙, 𝑧, 𝑡) = 𝑘𝜌3 𝑡 3 𝑧̂ , where 𝑘
The magnetic field at a point within the loop is, 𝐵
is a positive constant of appropriate dimensions. Neglecting the effects of any
current induced in the loop, the magnitude of the induced emf in the loop at time
𝑡 is

(A) 6𝜋𝑘𝑡 2 𝑅 5
5

(B) 5𝜋𝑘𝑡 2 𝑅 5
6

(C) 3𝜋𝑘𝑡 2 𝑅 5
2

(D) 𝜋𝑘𝑡 2 𝑅 5
2

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Q.25 For the given circuit, 𝑅 = 125 Ω, 𝑅𝐿 = 470 Ω, 𝑉𝑧 = 9 𝑉, and 𝐼𝑧𝑚𝑎𝑥 = 65 mA.
The minimum and maximum values of the input voltage (𝑉𝑖𝑚𝑖𝑛 and 𝑉𝑖𝑚𝑎𝑥 ) for
which the Zener diode will be in the ‘ON’ state are

(A) 𝑉𝑖𝑚𝑖𝑛 = 9.0 𝑉 and 𝑉𝑖𝑚𝑎𝑥 = 11.4 𝑉

(B) 𝑉𝑖𝑚𝑖𝑛 = 9.0 𝑉 and 𝑉𝑖𝑚𝑎𝑥 = 19.5 𝑉

(C) 𝑉𝑖𝑚𝑖𝑛 = 11.4 𝑉 and 𝑉𝑖𝑚𝑎𝑥 = 15.5 𝑉

(D) 𝑉𝑖𝑚𝑖𝑛 = 11.4 𝑉 and 𝑉𝑖𝑚𝑎𝑥 = 19.5 𝑉

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Q.26 A square laminar sheet with side 𝑎 and mass 𝑀, has mass per unit area given by
𝑥
𝜎(𝑥) = 𝜎0 [1 − 𝑎 ], (see figure). Moment of inertia of the sheet about 𝑦-axis is

(A) 𝑀𝑎2
2

(B) 𝑀𝑎2
4

(C) 𝑀𝑎2
6

(D) 𝑀𝑎2
12

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Q.27 A particle is subjected to two simple harmonic motions along the 𝑥 and 𝑦 axes,
described by 𝑥(𝑡) = 𝑎 sin(2𝜔𝑡 + 𝜋) and 𝑦(𝑡) = 2𝑎 sin(𝜔𝑡). The resultant
motion is given by

(A) 𝑥 2 𝑦 2
+ =1
𝑎2 4𝑎2

(B) 𝑥 2 + 𝑦 2 = 1

(C) 𝑥2
𝑦 2 = 𝑥 2 (1 − )
4𝑎2

(D) 𝑦2
𝑥 2 = 𝑦 2 (1 − )
4𝑎2

Q.28 For a certain thermodynamic system, the internal energy 𝑈 = 𝑃𝑉 and 𝑃 is
proportional to 𝑇 2 . The entropy of the system is proportional to

(A) 𝑈𝑉

(B)
𝑈
√
𝑉

(C)
𝑉
√
𝑈

(D) √𝑈𝑉

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Q.29 The dispersion relation for certain type of waves is given by 𝜔 = √𝑘 2 + 𝑎2 ,
where 𝑘 is the wave vector and 𝑎 is a constant. Which one of the following
sketches represents 𝑣𝑔 , the group velocity?

(A)

(B)

(C)

(D)

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Q.30 Consider a binary number with 𝑚 digits, where 𝑚 is an even number. This binary
number has alternating 1’s and 0’s, with digit 1 in the highest place value. The
decimal equivalent of this binary number is

(A) 2𝑚 − 1

(B) (2𝑚 − 1)
3

(C) (2𝑚+1 − 1)
3

(D) 2 𝑚
(2 − 1)
3

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Section B: Q.31 – Q.40 Carry TWO marks each.

Q.31 0 𝑎
Consider the 2 × 2 matrix 𝑀 = ( ), where 𝑎, 𝑏 > 0. Then,
𝑎 𝑏

(A) 𝑀 is a real symmetric matrix

(B) One of the eigenvalues of 𝑀 is greater than 𝑏

(C) One of the eigenvalues of 𝑀 is negative

(D) Product of eigenvalues of 𝑀 is 𝑏

Q.32 In the Compton scattering of electrons, by photons incident with wavelength λ,

(A) ∆𝜆
is independent of λ
𝜆

(B) ∆𝜆
increases with decreasing λ
𝜆

(C) there is no change in photon’s wavelength for all angles of deflection of the
photon

(D) ∆𝜆
increases with increasing angle of deflection of the photon
𝜆

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Q.33 The figure shows a section of the phase boundary separating the vapour (1) and
liquid (2) states of water in the P − 𝑇 plane. Here, 𝐶 is the critical point. 𝜇1 , 𝑣1
and 𝑠1 are the chemical potential, specific volume and specific entropy of the
vapour phase respectively, while 𝜇2 , 𝑣2 and 𝑠2 respectively denote the same for
the liquid phase. Then

(A) 𝜇1 = 𝜇2 along AB

(B) 𝑣1 = 𝑣2 along AB

(C) 𝑠1 = 𝑠2 along AB

(D) 𝑣1 = 𝑣2 at the point C

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Q.34 A particle is executing simple harmonic motion with time period 𝑇. Let 𝑥, 𝑣 and
𝑎 denote the displacement, velocity and acceleration of the particle, respectively,
at time t. Then,

(A) 𝑎𝑇
does not change with time
𝑥

(B) (𝑎𝑇 + 2𝜋𝑣) does not change with time

(C) 𝑥 and 𝑣 are related by an equation of a straight line

(D) 𝑣 and 𝑎 are related by an equation of an ellipse

Q.35 A linearly polarized light beam travels from origin to point A (1,0,0). At the point
A, the light is reflected by a mirror towards point B (1, −1, 0). A second mirror
located at point B then reflects the light towards point C (1, −1, 1). Let
𝑛̂(𝑥, 𝑦, 𝑧) represent the direction of polarization of light at (𝑥, 𝑦, 𝑧).

(A) If 𝑛̂(0, 0, 0) = 𝑦̂, then 𝑛̂(1, −1, 1) = 𝑥̂

(B) If 𝑛̂(0, 0, 0) = 𝑧̂ , then 𝑛̂(1, −1, 1) = 𝑦̂

(C) If 𝑛̂(0, 0, 0) = 𝑦̂, then 𝑛̂(1, −1, 1) = 𝑦̂

(D) If 𝑛̂(0, 0, 0) = 𝑧̂ , then 𝑛̂(1, −1, 1) = 𝑥̂

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Q.36 Let (𝑟, 𝜃) denote the polar coordinates of a particle moving in a plane. If 𝑟̂ and 𝜃̂
represent the corresponding unit vectors, then

(A) 𝑑𝑟̂
= 𝜃̂
𝑑𝜃

(B) 𝑑𝑟̂
= −𝜃̂
𝑑𝑟

(C) 𝑑𝜃̂
= −𝑟̂
𝑑𝜃

(D) 𝑑𝜃̂
= 𝑟̂
𝑑𝑟

Q.37 The electric field associated with an electromagnetic radiation is given by
𝐸 = 𝑎(1 + 𝑐𝑜𝑠𝜔1 𝑡)𝑐𝑜𝑠𝜔2 𝑡. Which of the following frequencies are present in
the field?

(A) 𝜔1

(B) 𝜔1 + 𝜔2

(C) |𝜔1 − 𝜔2 |

(D) 𝜔2

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Q.38 A string of length 𝐿 is stretched between two points 𝑥 = 0 and 𝑥 = 𝐿 and the
endpoints are rigidly clamped. Which of the following can represent the
displacement of the string from the equilibrium position?

(A) 𝜋𝑥
𝑥 cos ( )
𝐿

(B) 𝑥 sin (𝜋𝑥 )
𝐿

(C) 𝑥 (𝑥 − 1)
𝐿

(D) 𝑥 2
𝑥 ( − 1)
𝐿

Q.39 ̅̅̅̅ R+ Q𝑅̅ + 𝑃̅ 𝑄𝑅 + 𝑃𝑄𝑅 simplifies to
The Boolean expression 𝑌 = 𝑃𝑄

(A) 𝑃̅𝑅 + 𝑄

(B) 𝑃𝑅 + 𝑄̅

(C) 𝑃 + 𝑅

(D) 𝑄 + 𝑅

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Q.40 For an 𝑛-type silicon, an extrinsic semiconductor, the natural logarithm of

normalized conductivity (𝜎 ) is plotted as a function of inverse temperature.
Temperature interval-I corresponds to the intrinsic regime, interval-II
corresponds to saturation regime and interval-III corresponds to the freeze-out
regime, respectively. Then

(A) the magnitude of the slope of the curve in the temperature interval-I is
proportional to the bandgap, 𝐸𝑔

(B) the magnitude of the slope of the curve in the temperature interval-III is
proportional to the ionization energy of the donor, 𝐸𝑑

(C) in the temperature interval-II, the carrier density in the conduction band is equal
to the density of donors

(D) in the temperature interval-III, all the donor levels are ionized

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JAM 2022 PHYSICS - PH

Section C: Q.41 – Q.50 Carry ONE mark each.

Q.41 The integral ∬(𝑥 2 + 𝑦 2 )𝑑𝑥𝑑𝑦 over the area of a disk of radius 2 in the 𝑥𝑦
plane is ___𝜋.

Q.42 For the given operational amplifier circuit 𝑅1 = 120 Ω, 𝑅2 = 1.5 kΩ and
𝑉𝑠 = 0.6 V, then the output current 𝐼0 is ___________ mA.

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Q.43 For an ideal gas, AB and CD are two isothermals at temperatures 𝑇1 and 𝑇2 (𝑇1 >
𝑇2 ), respectively. AD and BC represent two adiabatic paths as shown in figure.
Let 𝑉𝐴 , 𝑉𝐵 , 𝑉𝐶 and 𝑉𝐷 be the volumes of the gas at A, B, C and D respectively. If
𝑉𝐶 𝑉
= 2 , then 𝑉𝐷 = __________.
𝑉𝐵 𝐴

Q.44 A satellite is revolving around the Earth in a closed orbit. The height of the
satellite above Earth’s surface at perigee and apogee are 2500 km and 4500 km,
respectively. Consider the radius of the Earth to be 6500 km. The eccentricity of
the satellite’s orbit is ____ (Round off to 1 decimal place).

Q.45 Three masses 𝑚1 = 1, 𝑚2 = 2 and 𝑚3 = 3 are located on the 𝑥-axis such that
their center of mass is at 𝑥 = 1 . Another mass 𝑚4 = 4 is placed at 𝑥0 and the
new center of mass is at 𝑥 = 3. The value of 𝑥0 is _____.

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JAM 2022 PHYSICS - PH

Q.46 A normal human eye can distinguish two objects separated by 0.35 𝑚 when
viewed from a distance of 1.0 𝑘𝑚. The angular resolution of eye is
________seconds (Round off to the nearest integer).

Q.47 A rod with a proper length of 3 m moves along 𝑥-axis, making an angle of 300
𝑐
with respect to the 𝑥-axis. If its speed is 2 m/s, where c is the speed of light, the

change in length due to Lorentz contraction is ____m (Round off to 2 decimal
places).

[Use 𝑐 = 3 × 108 𝑚/𝑠]

Q.48 Consider the Bohr model of hydrogen atom. The speed of an electron in the
second orbit (𝑛 = 2) is ______× 106 𝑚/𝑠 (Round off to 2 decimal places).

[Use ℎ = 6.63 × 10−34 𝐽𝑠, 𝑒 = 1.6 × 10−19 𝐶, 𝜖0 = 8.85 × 10−12 𝐶 2 𝑚2 /𝑁]

Q.49 Consider a unit circle 𝐶 in the 𝑥𝑦 plane with center at the origin. The line integral
of the vector field, 𝐹 (𝑥, 𝑦, 𝑧) = −2𝑦𝑥̂ − 3𝑧𝑦̂ + 𝑥𝑧̂ , taken anticlockwise over 𝐶
is _____ 𝜋.

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Q.50 Consider a p-n junction at 𝑇 = 300 𝐾. The saturation current density at reverse
bias is −20 𝜇𝐴/𝑐𝑚2. For this device, a current density of magnitude
10 𝜇𝐴/𝑐𝑚2 is realized with a forward bias voltage, 𝑉𝐹 . The same magnitude of
current density can also be realized with a reverse bias voltage, 𝑉𝑅 . The value of
|𝑉𝐹 /𝑉𝑅 | is _______ (Round off to 2 decimal places).

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Section C: Q.51 – Q.60 Carry TWO marks each.

Q.51 Consider the second order ordinary differential equation, 𝑦 ′′ + 4𝑦 ′ + 5𝑦 = 0. If
𝑦(0) = 0 and 𝑦′(0) = 1, then the value of 𝑦(𝜋/2) is __________ (Round off
to 3 decimal places).

Q.52 A box contains a mixture of two different ideal monoatomic gases, 1 and 2, in
equilibrium at temperature 𝑇. Both gases are present in equal proportions. The
atomic mass for gas 1 is 𝑚, while the same for gas 2 is 2𝑚 . If the rms speed of

𝐵 𝑘 𝑇
a gas molecule selected at random is 𝑣𝑟𝑚𝑠 = 𝑥√ 𝑚 , then 𝑥 is __________

(Round off to 2 decimal places).

Q.53 A hot body with constant heat capacity 800 𝐽/𝐾 at temperature 925 𝐾 is dropped
gently into a vessel containing 1 𝑘𝑔 of water at temperature 300 𝐾 and the
combined system is allowed to reach equilibrium. The change in the total entropy
Δ𝑆 is _________ 𝐽/𝐾 (Round off to 1 decimal place).

[Take the specific heat capacity of water to be 4200 𝐽/𝑘𝑔 𝐾. Neglect any loss of
heat to the vessel and air and change in the volume of water.]

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Q.54 Consider an electron with mass 𝑚 and energy 𝐸 moving along the 𝑥-axis towards
a finite step potential of height 𝑈0 as shown in the figure. In region 𝟏 (𝑥 < 0), the
momentum of the electron is 𝑝1 = √2𝑚𝐸. The reflection coefficient at the barrier
𝑝 −𝑝 2
is given by 𝑅 = (𝑝1 +𝑝2 ) , where 𝑝2 is the momentum in region 𝟐. If, in the limit
1 2

𝑈2
𝐸 ≫ 𝑈0 , 𝑅 ≈ 𝑛𝐸02 , then the integer 𝑛 is ___.

Q.55 A current density for a fluid flow is given by,

8𝑒 𝑡
𝐽 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑥̂ .
(1+𝑥 2 +𝑦2 +𝑧 2 )

At time 𝑡 = 0, the mass density 𝜌(𝑥, 𝑦, 𝑧, 0) = 1.
Using the equation of continuity, 𝜌(1,1,1,1) is found to be _______ (Round off
to 2 decimal places).

Q.56 The work done in moving a −5 𝜇𝐶 charge in an electric field
𝜋
𝐸⃗ = (8𝑟 sin 𝜃 𝑟̂ + 4𝑟 cos 𝜃 𝜃̂) 𝑉/𝑚, from a point 𝐴(𝑟, 𝜃) = (10, 6 ) to a point
𝜋
𝐵(𝑟, 𝜃) = (10, 2 ), is ______ m𝐽.

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Q.57 A pipe of 1 𝑚 length is closed at one end. The air column in the pipe resonates at
its fundamental frequency of 400 𝐻𝑧. The number of nodes in the sound wave
formed in the pipe is _____.

[Speed of sound = 320 𝑚/𝑠]

Q.58 The critical angle of a crystal is 30∘ . Its Brewster angle is ______ degrees (Round
off to the nearest integer).

Q.59 In an LCR series circuit, a non-inductive resistor of 150 Ω, a coil of 0.2 𝐻
inductance and negligible resistance, and a 30 µ𝐹 capacitor are connected across
an ac power source of 220 𝑉, 50 Hz. The power loss across the resistor is ____W
(Round off to 2 decimal places).

Q.60 A charge q is uniformly distributed over the volume of a dielectric sphere of
radius a. If the dielectric constant 𝜖𝑟 = 2, then the ratio of the electrostatic energy
stored inside the sphere to that stored outside is_______ (Round off to 1 decimal
place).

END OF THE QUESTION PAPER

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