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GATE (Graduate Aptitude Test in Engineering) Physics is an online examination conducted to get admission in M-Tech (Masters in Technology), ME (Master of Engineering ) and PhD at IITs (Indian Institutes of Technology), NITs (National Institutes of Technology), IIITs (Indian Institutes of Information Technology) and other universities in India. GATE exam is jointly conducted by IIT Bombay, IIT Delhi, IIT Guwahati, IIT Kanpur, IIT Kharagpur, IIT Madras, IIT Roorkee and Indian Institute for Science, Bangalore (IISC Bangalore). The exam is divided into two different parts General Aptitude (GA) and Candidate’s Selected Subject (Core Subjects). GATE Physics online mock tests help candidates in preparing for the GATE exam. By attempting GATE online mock tests candidates get familiar with the pattern of actual question paper. BrPaper provides all GATE question papers completely free. Here you can practice free online mock tests of complete exam of GATE Physics. We advise candidates to solve the online GATE Physics mock tests to get good marks in the exam. Start practicing PhysicsGATE 2019 Mock tests here online for free

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Category: GATE Physics 2019 General Aptitude

Question 1:

The fishermen, _______ the flood victims owed their lives, were rewarded by the government.

whom

to which

to whom

that

Category: GATE Physics 2019 General Aptitude

Question 2:

Some students were not involved in the strike. If the above statement is true, which of the following conclusions is/are logically necessary? 1. Some who were involved in the strike were students. 2. No student was involved in the strike. 3. At least one student was involved in the strike. 4. Some who were not involved in the strike were students.

1 and 2

2 and 3

Category: GATE Physics 2019 General Aptitude

Question 3:

The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is ______.

17.1

21.0

33.1

72.8

Category: GATE Physics 2019 General Aptitude

Question 4:

Five numbers 10, 7, 5, 4 and 2 are to be arranged in a sequence from left to right following the directions given below: 1. No two odd or even numbers are next to each other. 2. The second number from the left is exactly half of the left-most number. 3. The middle number is exactly twice the right-most number. Which is the second number from the right?

Category: GATE Physics 2019 General Aptitude

Question 5:

Until Iran came along, India had never been ________________ in kabaddi.

defeated

defeating

defeat

defeatist

Category: GATE Physics 2019 General Aptitude

Question 6:

Since the last one year, after a 125 basis point reduction in repo rate by the Reserve Bank of India, banking institutions have been making a demand to reduce interest rates on small saving schemes. Finally, the government announced yesterday a reduction in interest rates on small saving schemes to bring them on par with fixed deposit interest rates. Which one of the following statements can be inferred from the given passage?

A	Whenever the Reserve Bank of India reduces the repo rate, the interest rates on small saving schemes are also reduced
B	Interest rates on small saving schemes are always maintained on par with fixed deposit interest rates
C	The government sometimes takes into consideration the demands of banking institutions before reducing the interest rates on small saving schemes
D	A reduction in interest rates on small saving schemes follow only after a reduction in repo rate by the Reserve Bank of India

Category: GATE Physics 2019 General Aptitude

Question 7:

In a country of 1400 million population, 70% own mobile phones. Among the mobile phone owners, only 294 million access the Internet. Among these Internet users, only half buy goods from e-commerce portals. What is the percentage of these buyers in the country?

10.50

14.70

15.00

50.00

Category: GATE Physics 2019 General Aptitude

Question 8:

The nomenclature of Hindustani music has changed over the centuries. Since the medieval period dhrupad styles were identified as baanis. Terms like gayaki and baaj were used to refer to vocal and instrumental styles, respectively. With the institutionalization of music education the term gharana became acceptable. Gharana originally referred to hereditary musicians from a particular lineage, including disciples and grand disciples. Which one of the following pairings is NOT correct?

dhrupad, baani

gayaki, vocal

baaj, institution

gharana, lineage

Category: GATE Physics 2019 General Aptitude

Question 9:

Two trains started at 7AM from the same point. The first train travelled north at a speed of 80km/h and the second train travelled south at a speed of 100 km/h. The time at which they were 540 km apart is ______ AM.

11.30

Category: GATE Physics 2019 General Aptitude

Question 10:

“I read somewhere that in ancient times the prestige of a kingdom depended upon the number of taxes that it was able to levy on its people. It was very much like the prestige of a head-hunter in his own community.” Based on the paragraph above, the prestige of a head-hunter depended upon ___________

the prestige of the kingdom

the prestige of the heads

the number of taxes he could levy

the number of heads he could gather

Category: GATE Physics 2019

Question 11:

The relative magnetic permeability of a type-I superconductor is

A	0	B	-1
C	2π	D	$\frac{1}{4 π}$

Category: GATE Physics 2019

Question 12:

Considering baryon number and lepton number conservation laws, which of the following processes is/are allowed?

(i) ρ \to π^{0} + e^{+} + v_{e} (ii) e^{+} + v_{e} \to μ^{+} + v_{μ}

both (i) and (ii)

only (i)

only (ii)

neither (i) nor (ii)

Category: GATE Physics 2019

Question 13:

For the following circuit, what is the magnitude of

V_{o u t} i f V_{i n} = 1.5 V

0.015 V

0.15 V

15 V

150 V

Category: GATE Physics 2019

Question 14:

For the differential equation

\frac{d^{2} y}{d x^{2}} - n (n + 1) \frac{y}{x^{2}} = 0,

, where 𝑛 is a constant, the product of its two independent solutions is

A	$\frac{1}{x}$	B	x
C	$x^{n}$	D	$\frac{1}{x^{n + 1}}$

Category: GATE Physics 2019

Question 15:

Consider a one-dimensional gas of 𝑁 non-interacting particles of mass 𝑚 with the Hamiltonian for a single particle given by,

H = \frac{ρ^{2}}{2 m} + \frac{1}{2} m ω^{2} (x^{2} + 2 x)

The high temperature specific heat in units of 𝑅 =

N k_{B}

(

k_{B}

is the Boltzmann constant) is

1.5

2.5

Category: GATE Physics 2019

Question 16:

An electric field

\vec{E} = E_{0} \hat{z}

is applied to a Hydrogen atom in 𝑛 = 2 excited state. Ignoring spin, the 𝑛 = 2 state is fourfold degenerate, which in the |𝑙, 𝑚⟩ basis are given by |0,0〉, |1,1⟩, |1,0⟩ and |1, −1⟩. If 𝐻′ is the interaction Hamiltonian corresponding to the applied electric field, which of the following matrix elements is nonzero?

A	⟨0,0\|𝐻′\|0,0⟩	B	⟨0,0\|𝐻′\|1,1⟩
C	⟨0,0\|𝐻′\|1,0⟩	D	⟨0,0\|𝐻′\|1, −1⟩

Category: GATE Physics 2019

Question 17:

A large number 𝑁 of ideal bosons, each of mass m, are trapped in a three-dimensional potential

V (r) = \frac{m ω^{2} r^{2}}{2} .

. The bosonic system is kept at temperature 𝑇 which is much lower than the Bose-Einstein condensation temperature

T_{C}

. The chemical potential (𝜇) satisfies

A	$μ \leq \frac{3}{2} ℏ ω$	B	$2 ℏ ω > μ > \frac{3}{2} ℏ ω$
C	$3 ℏ ω > μ > 2 ℏ ω$	D	$μ = 3 ℏ ω$

Category: GATE Physics 2019

Question 18:

During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix

(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 1 \\ - 1 & 0 & 0 \end{matrix})

, the unit vector along the axis of rotation is

A	$\frac{1}{3} (2 \hat{i} - \hat{j} + 2 \hat{k})$	B	$\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} - \hat{k})$
C	$\frac{1}{\sqrt{3}} (\hat{i} - \hat{j} - \hat{k})$	D	$\frac{1}{3} (2 \hat{i} + 2 \hat{j} - \hat{k})$

Category: GATE Physics 2019

Question 19:

For a spin

\frac{1}{2}

particle, let | ↑⟩ and | ↓⟩ denote its spin up and spin down states, respectively. If |𝑎⟩ =

\frac{1}{\sqrt{2}}

( | ↑⟩ | ↓⟩ + | ↓⟩ | ↑⟩) and |𝑏⟩ =

\frac{1}{\sqrt{2}}

( | ↑⟩ | ↓⟩ − | ↓⟩ | ↑⟩) are composite states of two such particles, which of the following statements is true for their total spin S ?

A	S=1 for \|𝑎⟩, and \|𝑏⟩ is not an eigenstate of the operator ${\hat{S}}^{2}$
B	\|𝑎⟩ is not an eigenstate of the operator ${\hat{S}}^{2}$ , and S=0 for \|𝑏⟩
C	S=0 for \|𝑎⟩, and S=1 for \|𝑏⟩
D	S=1 for \|𝑎⟩, and S=0 for \|𝑏⟩

Category: GATE Physics 2019

Question 20:

Consider a transformation from one set of generalized coordinate and momentum (𝑞, 𝑝) to another set (𝑄, 𝑃) denoted by,

Q = p q^{s}; P = q^{r}

where 𝑠 and 𝑟 are constants. The transformation is canonical if

A	𝑠 = 0 and 𝑟 = 1	B	𝑠 = 2 and 𝑟 = −1
C	𝑠 = 0 and 𝑟 = −1	D	𝑠 = 2 and 𝑟 = 1

Category: GATE Physics 2019

Question 21:

In order to estimate the specific heat of phonons, the appropriate method to apply would be

A	Einstein model for acoustic phonons and Debye model for optical phonons
B	Einstein model for optical phonons and Debye model for acoustic phonons
C	Einstein model for both optical and acoustic phonons
D	Debye model for both optical and acoustic phonons

Category: GATE Physics 2019

Question 22:

The pole of the function 𝑓(𝑧) = cot 𝑧 at 𝑧 = 0 is

a removable singularity

an essential singularity

a simple pole

a second order pole

Category: GATE Physics 2019

Question 23:

A massive particle X in free space decays spontaneously into two photons. Which of the following statements is true for X?

A	X is charged	B	Spin of X must be greater than or equal to 2
C	X is a boson	D	X must be a baryon

Category: GATE Physics 2019

Question 24:

The electric field of an electromagnetic wave is given by

\vec{E} = 3 \sin (k z - ω t) \hat{x} + 4 \cos (k z - ω t) \hat{y} .

The wave is

A	linearly polarized at an angle $\tan^{- 1} (\frac{4}{3})$ from the x-axis
B	linearly polarized at an angle $\tan^{- 1} (\frac{3}{4})$ from the x-axis
C	elliptically polarized in clockwise direction when seen travelling towards the observer
D	elliptically polarized in counter-clockwise direction when seen travelling towards the observer

Category: GATE Physics 2019

Question 25:

The nuclear spin and parity of

C_{20}^{40} a

in its ground state is

A	$0^{+}$	B	$0^{-}$
C	$1^{+}$	D	$1^{-}$

Category: GATE Physics 2019

Question 26:

An infinitely long thin cylindrical shell has its axis coinciding with the z-axis. It carries a surface charge density

σ_{0}

cos𝜙, where 𝜙 is the polar angle and

σ_{0}

is a constant. The magnitude of the electric field inside the cylinder is

A	0	B	$\frac{σ_{0}}{2 \in_{0}}$
C	$\frac{σ_{0}}{3 \in_{0}}$	D	$\frac{σ_{0}}{4 \in_{0}}$

Category: GATE Physics 2019

Question 27:

Consider a three-dimensional crystal of 𝑁 inert gas atoms. The total energy is given by

U (R) = [p {(\frac{σ}{R})}^{12} - q {(\frac{σ}{R})}^{6}],

, where 𝑝 = 12.13, 𝑞 = 14.45, and 𝑅 is the nearest neighbour distance between two atoms. The two constants, 𝜖 and 𝑅, have the dimensions of energy and length, respectively. The equilibrium separation between two nearest neighbour atoms in units of 𝜎 (rounded off to two decimal places) is ___________

1.06 to 1.11

1.07 to 1.11

1.08 to 1.11

1.09 to 1.11

Category: GATE Physics 2019

Question 28:

The energy-wavevector (𝐸-𝑘) dispersion relation for a particle in two dimensions is 𝐸 = 𝐶𝑘, where 𝐶 is a constant. If its density of states 𝐷(𝐸) is proportional to

E^{p}

then the value of 𝑝 is ______________

0 to 0

0 to 1

-1 to 1

1 to 1

Category: GATE Physics 2019

Question 29:

A circular loop made of a thin wire has radius 2 cm and resistance 2 Ω. It is placed perpendicular to a uniform magnetic field of magnitude

|{\vec{B}}_{0}|

= 0.01

Tesla. At time 𝑡 = 0 the field starts decaying as

\vec{B} = {\vec{B}}_{0} e^{- t / t_{0}}, w h e r e t_{0} = 1 s .

The total charge that passes through a cross section of the wire during the decay is Q. The value of Q in 𝜇𝐶 (rounded off to two decimal places) is________________

6.26 to 6.30

6.27 to 6.30

6.28 to 6.30

6.29 to 6.30

Category: GATE Physics 2019

Question 30:

The electric field of an electromagnetic wave in vacuum is given by

\vec{E} = E_{0} \cos (3 y + 4 z - 1.5 x 10^{9} t) \hat{x} .

The wave is reflected from the 𝑧 = 0 surface. If the pressure exerted on the surface is

α \in_{0} E_{0}^{2},

the value of 𝛼 (rounded off to one decimal place) is ___________

0.2 to 0.2

0.4 to 0.4

0.6 to 0.6

0.8 to 0.8

Category: GATE Physics 2019

Question 31:

The Hamiltonian for a quantum harmonic oscillator of mass 𝑚 in three dimensions is

H = \frac{p^{2}}{2 m} + \frac{1}{2} m ω^{2} r^{2}

where 𝜔 is the angular frequency. The expectation value of

r^{2}

in the first excited state of the oscillator in units of

\frac{ℏ}{m ω}

(rounded off to one decimal place) is __________

2.0 to 2.0

2.0 to 2.5

2.5 to 2.5

2.5 to 3.0

Category: GATE Physics 2019

Question 32:

The Hamiltonian for a particle of mass 𝑚 is 𝐻

= \frac{p^{2}}{2 m} + k q t

where 𝑞 and 𝑝 are the generalized coordinate and momentum, respectively, 𝑡 is time and 𝑘 is a constant. For the initial codition,

q = 0 a n d p = 0 a t t = 0, q (t)

\propto t^{α}

. The value of α is _______

1 to 1

2 to 2

3 to 3

4 to 4

Category: GATE Physics 2019

Question 33:

At temperature T Kelvin (K), the value of the Fermi function at an energy 0.5 eV above the Fermi energy is 0.01. Then T, to the nearest integer, is_____________

(k_{B} = 8, 62 x 10^{- 5} e V / K)

1260 to 1266

1262 to 1266

1264 to 1266

1266 to 1266

Category: GATE Physics 2019

Question 34:

Let

| {��}_{1} 〉 = (\begin{matrix} 1 \\ 0 \end{matrix}), | {��}_{2} 〉 = (\begin{matrix} 0 \\ 1 \end{matrix})

represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of

| {��}_{1} ⟩ a n d | {��}_{2}

is given by a density matrix that is defined as

�� \equiv c_{1} | {��}_{1} 〉 〈 {��}_{1} | + c_{2} | {��}_{2} 〉 〈 {��}_{2} | . I f c_{1} = 0.4 a n d {��}_{2} = 0.6,

= 0.6, the matrix element

p_{22}

(rounded off to one decimal place) is _________

0.0 to 0.0

0.3 to 0.3

0.6 to 0.6

0.9 to 0.9

Category: GATE Physics 2019

Question 35:

A conventional type-I superconductor has a critical temperature of 4.7 K at zero magnetic field and a critical magnetic field of 0.3 Tesla at 0 K. The critical field in Tesla at 2 K (rounded off to three decimal places) is ____________

0.241 to 0.248

0.242 to 0.248

0.243 to 0.248

0.244 to 0.248

Category: GATE Physics 2019

Question 36:

Consider the following Boolean expression:

(A + B) [A (B + C)] + A (B + \bar{C})

It can be represented by a single three-input logic gate. Identify the gate.

AND

XOR

NAND

Category: GATE Physics 2019

Question 37:

The value of the integral

\int_{- \infty}^{\infty} \frac{\cos (k x)}{x^{2} + α^{2}} d x, w h e r e k > 0 a n d α > 0, i s

A	$\frac{π}{α} e^{- k α}$	B	$\frac{2 π}{α} e^{- k α}$
C	$\frac{π}{2 α} e^{- k α}$	D	$\frac{3 π}{2 α} e^{- k α}$

Category: GATE Physics 2019

Question 38:

The wave function 𝜓(𝑥) of a particle is as shown below

Here 𝐾 is a constant, and 𝑎 > 𝑑. The position uncertainty 〈Δ𝑥〉 of the particle is

A	$\sqrt{\frac{α^{2} + 3 d^{2}}{12}}$	B	$\sqrt{\frac{3 α^{2} + d^{2}}{12}}$
C	$\sqrt{\frac{d^{2}}{6}}$	D	$\sqrt{\frac{d^{2}}{24}}$

Category: GATE Physics 2019

Question 39:

A solid cylinder of radius 𝑅 has total charge 𝑄 distributed uniformly over its volume. It is rotating about its axis with angular speed 𝜔. The magnitude of the total magnetic moment of the cylinder is

A	$Q R^{2} ω$	B	$\frac{1}{2} Q R^{2} ω$
C	$\frac{1}{4} Q R^{2} ω$	D	$\frac{1}{8} Q R^{2} ω$

Category: GATE Physics 2019

Question 40:

Consider the motion of a particle along the x-axis in a potential 𝑉(𝑥) = 𝐹|𝑥|. Its ground state energy

E_{0}

is estimated using the uncertainty principle. Then

E_{0}

is proportional to

A	$F^{\frac{1}{3}}$	B	$F^{\frac{1}{2}}$
C	$F^{\frac{2}{5}}$	D	$F^{\frac{2}{3}}$

Category: GATE Physics 2019

Question 41:

A 3-bit analog-to-digital converter is designed to digitize analog signals ranging from 0 V to 10 V. For this converter, the binary output corresponding to an input of 6 V is

011

101

100

010

Category: GATE Physics 2019

Question 42:

The Hamiltonian operator for a two-level quantum system is 𝐻 =

(\begin{matrix} E_{1} & 0 \\ 0 & E_{2} \end{matrix})

. If the state of the system at 𝑡 = 0 is given by |𝜓 (0)〉 =

\frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix})

then

| ⟨ ψ (0) | ψ (𝑡) ⟩ |^{2}

at a later time 𝑡 is

A	$\frac{1}{2} (1 + e^{- (E_{1} - E_{2}) t / ℏ})$	B	$\frac{1}{2} (1 - e^{- (E_{1} - E_{2}) t / ℏ})$
C	$\frac{1}{2} (1 + \cos [(E_{1} - E_{2}) t / ℏ])$	D	$\frac{1}{2} (1 - \cos [(E_{1} - E_{2}) t / ℏ])$

Category: GATE Physics 2019

Question 43:

A particle of mass m moves in a lattice along the x-axis in a periodic potential 𝑉(𝑥) = 𝑉(𝑥 + 𝑑) with periodicity d. The corresponding Brillouin zone extends from

- k_{0} t o k_{0}

with these two k-points being equivalent. If a weak force F in the x-direction is applied to the particle, it starts a periodic motion with time period T. Using the equation of motion

F = \frac{d p c r y s t a l}{d t}

for a particle moving in a band, where 𝑝crystal is the crystal momentum of the particle, the period T is found to be (ℎ is Planck constant)

A	$\sqrt{\frac{2 m d}{F}}$	B	$2 \sqrt{\frac{2 m d}{F}}$
C	$\frac{2 h}{F d}$	D	$\frac{h}{F d}$

Category: GATE Physics 2019

Question 44:

Consider a potential barrier 𝑉(𝑥) of the form:

where

V_{0}

is a constant. For particles of energy 𝐸 <

V_{0}

incident on this barrier from the left, which of the following schematic diagrams best represents the probability density

| ψ (𝑥) |^{2}

as a function of 𝑥.

Category: GATE Physics 2019

Question 45:

The spin-orbit interaction term of an electron moving in a central field is written as

f (r) \vec{l} \cdot \vec{s},

, where r is the radial distance of the electron from the origin. If an electron moves inside a uniformly charged sphere, then

A	$f (r) = c o n s \tan t$	B	$f (r) \propto r^{- 1}$
C	$f (r) \propto r^{- 2}$	D	$f (r) \propto r^{- 3}$

Category: GATE Physics 2019

Question 46:

For the following circuit, the correct logic values for the entries

X_{2} a n d Y_{2}

in the truth table are

1 and 0

0 and 0

0 and 1

1 and 1

Category: GATE Physics 2019

Question 47:

In a set of 𝑁 successive polarizers, the

m^{t h}

polarizer makes an angle

(\frac{m π}{2 N})

with the vertical. A vertically polarized light beam of intensity 𝐼0 is incident on two such sets with

N = N_{1} a n d N = N_{2}

, where

N_{2} > N_{1}

. Let the intensity of light beams coming out be 𝐼(

N_{1}

) and 𝐼(

N_{2}

) , respectively. Which of the following statements is correct about the two outgoing beams?

A	$I (N_{2}) > I (N_{1})$ ; the polarization in each case is vertical
B	$I (N_{2}) < I (N_{1})$ ; the polarization in each case is vertical
C	$I (N_{2}) > I (N_{1})$ ; the polarization in each case is horizontal
D	$I (N_{2}) < I (N_{1})$ ; the polarization in each case is horizontal

Category: GATE Physics 2019

Question 48:

A ball bouncing off a rigid floor is described by the potential energy function

V (x) = m g x f o r x > 0 = \infty f o r x \leq 0

Which of the following schematic diagrams best represents the phase space plot of the ball?

Category: GATE Physics 2019

Question 49:

An infinitely long wire parallel to the x-axis is kept at 𝑧 = 𝑑 and carries a current I in the positive x direction above a superconductor filling the region 𝑧 ≤ 0 (see figure). The magnetic field

\vec{B}

inside the superconductor is zero so that the field just outside the superconductor is parallel to its surface. The magnetic field due to this configuration at a point (𝑥, 𝑦, 𝑧 > 0) is

A	$(\frac{μ_{0} I}{2 π}) \frac{- (z - d) \hat{j} + y \hat{k}}{[y^{2} + {(z - d)}^{2}]}$
B	$(\frac{μ_{0} I}{2 π}) [\frac{- (z - d) \hat{j} + y \hat{k}}{y^{2} + {(z - d)}^{2}} + \frac{(z + d) \hat{j} - y \hat{k}}{y^{2} + {(z + d)}^{2}}]$
C	$(\frac{μ_{0} I}{2 π}) [\frac{- (z - d) \hat{j} + y \hat{k}}{y^{2} + {(z - d)}^{2}} - \frac{(z + d) \hat{j} - y \hat{k}}{y^{2} + {(z + d)}^{2}}]$
D	$(\frac{μ_{0} I}{2 π}) [\frac{y \hat{j} (z - d) \hat{k}}{y^{2} + {(z - d)}^{2}} + \frac{y \hat{j} - (z + d) \hat{k}}{y^{2} + {(z + d)}^{2}}]$

Category: GATE Physics 2019

Question 50:

The vector potential inside a long solenoid, with n turns per unit length and carrying current 𝐼, written in cylindrical coordinates is

\vec{A}

(𝑠,𝜙, 𝑧) =

\frac{μ_{0} n I}{2}

𝑠 𝜙̂. If the term

\frac{μ_{0} n I}{2}

s(𝛼 cos𝜙 𝜙̂ + 𝛽 sin𝜙 𝑠̂), where 𝛼 ≠ 0 , 𝛽 ≠ 0, is added to

\vec{A}

(𝑠,𝜙, 𝑧), the magnetic field remains the same if

A	$α = β$	B	$α = - β$
C	$α = 2 β$	D	$α = \frac{β}{2}$

[\begin{matrix} U s e f u l F o r m u l a e : \vec{\nabla} = \frac{�� t}{�� s} \hat{s} + \frac{1}{s} \frac{�� t}{�� ϕ} \hat{ϕ} + \frac{�� t}{�� z} \hat{z}; \\ \vec{\nabla} x \vec{v} = (\frac{1}{s} \frac{�� v_{z}}{�� ϕ} - \frac{�� v_{ϕ}}{�� z}) \hat{s} + (\frac{�� v_{s}}{�� z} - \frac{�� v_{z}}{�� s}) \hat{ϕ} + \frac{1}{s} (\frac{�� (s v_{ϕ}}{�� s} - \frac{�� v_{s}}{�� ϕ}) \hat{z} \end{matrix}]

Category: GATE Physics 2019

Question 51:

Low energy collision (s-wave scattering) of pion

(π^{+})

with deuteron (d) results in the production of two protons (

π^{+} +

d → p + p). The relative orbital angular momentum (in units of ℏ) of the resulting two-proton system for this reaction is

Category: GATE Physics 2019

Question 52:

Consider the Hamiltonian

H (q, p) = \frac{α p^{2} q^{4}}{2} + \frac{β}{q^{2}},

, where 𝛼 and 𝛽 are parameters with appropriate dimensions, and 𝑞 and 𝑝 are the generalized coordinate and momentum, respectively. The corresponding Lagrangian 𝐿(𝑞, 𝑞̇) is

A	$\frac{1}{2 α} \frac{q^{2}}{q^{4}} - \frac{β}{q^{2}}$	B	$\frac{1}{α} \frac{q^{2}}{q^{4}} - \frac{β}{q^{2}}$
C	$\frac{1}{α} \frac{q^{2}}{q^{4}} + \frac{β}{q^{2}}$	D	$- \frac{1}{2 α} \frac{q^{2}}{q^{4}} + \frac{β}{q^{2}}$

Category: GATE Physics 2019

Question 53:

For a given load resistance

R_{L}

= 4.7 ohm, the power transfer efficiencies

(η = \frac{P_{l o a d}}{P_{t o t a l}})

of a dc voltage source and a dc current source with internal resistances

R_{1} a n d R_{2}

, respectively, are equal. The product

R_{1} R_{2}

in units of

o h m^{2}

(rounded off to one decimal place) is

22.0 to 22.2

22.0 to 22.4

22.0 to 22.6

22.0 to 22.8

Category: GATE Physics 2019

Question 54:

The ground state electronic configuration of the rare-earth ion

(N d^{3 +}) i s [P d] 4 f^{3} 5 s^{2} 5 p^{6} .

Assuming LS coupling, the Land𝑒́ 𝑔-factor of this ion is

\frac{8}{11}

. The effective magnetic moment in units of Bohr magneton

μ_{B}

(rounded off to two decimal places) is

3.60 to 3.63

3.61 to 3.63

3.62 to 3.63

3.63 to 3.63

Category: GATE Physics 2019

Question 55:

A projectile of mass 1 kg is launched at an angle of 30° from the horizontal direction at 𝑡 = 0 and takes time 𝑇 before hitting the ground. If its initial speed is 10

m s^{- 1}

, the value of the action integral for the entire flight in the units of

K g m^{2} s^{- 1}

(rounded off to one decimal place) is _____________ [ Take g =10

m s^{- 2}

]

33.0 to 33.2

33.2 to 33.4

33.4 to 33.6

33.6 to 33.8

Category: GATE Physics 2019

Question 56:

Let 𝜃 be a variable in the range − 𝜋 ≤ 𝜃 < 𝜋. Now consider a function

ψ (θ) = 1 f o r - π / 2 \leq θ < π / 2 = 0 o t h e r w i s e .

If its Fourier-series is written as

ψ (θ) = \sum_{m = - \infty}^{\infty} C_{m} e^{- i m θ}

, then the value of

{|C_{3}|}^{2}

(rounded off to three decimal places) is _________________

0.010 to 0.010

0.010 to 0.011

0.010 to 0.012

0.010 to 0.013

Category: GATE Physics 2019

Question 57:

Two spaceships A and B, each of the same rest length 𝐿, are moving in the same direction with speeds

\frac{4 c}{5}

and

\frac{3 c}{5},

respectively, where 𝑐 is the speed of light. As measured by B, the time taken by A to completely overtake B [see figure below] in units of 𝐿/𝑐 (to the nearest integer) is _______________

1 to 1

3 to 3

5 to 5

7 to 7

Category: GATE Physics 2019

Question 58:

A radioactive element X has a half-life of 30 hours. It decays via alpha, beta and gamma emissions with the branching ratio for beta decay being 0.75. The partial half-life for beta decay in unit of hours is ________________

10 to 10

20 to 20

30 to 30

40 to 40

Category: GATE Physics 2019

Question 59:

In a thermally insulated container, 0.01 kg of ice at 273 K is mixed with 0.1 kg of water at 300 K. Neglecting the specific heat of the container, the change in the entropy of the system in J/K on attaining thermal equilibrium (rounded off to two decimal places) is ___________ (Specific heat of water is 4.2 kJ/kg-K and the latent heat of ice is 335 kJ/kg).

1.00 to 1.02

1.02 to 1.04

1.04 to 1.06

1.06 to 1.08

Category: GATE Physics 2019

Question 60:

Consider a system of three charges as shown in the figure below:

For 𝑟 = 10 m; 𝜃 = 60 degrees; 𝑞 =

10^{- 6}

Coulomb, and 𝑑 =

10^{- 3}

m, the electric dipole potential in volts (rounded off to three decimal places) at a point (𝑟, 𝜃) is __________________

[U s e : \frac{1}{4 π \in_{0}} = 9 x 10^{9} \frac{N m^{2}}{C^{2}}]

0.040 to 0.042

0.042 to 0.044

0.044 to 0.046

0.046 to 0.048

Category: GATE Physics 2019

Question 61:

Consider two systems A and B each having two distinguishable particles. In both the systems, each particle can exist in states with energies 0, 1, 2 and 3 units with equal probability. The total energy of the combined system is 5 units. Assuming that the system A has energy 3 units and the system B has energy 2 units, the entropy of the combined system is

K_{B}

ln 𝜆. The value of 𝜆 is________________

12 to 12

13 to 13

14 to 14

15 to 15

Category: GATE Physics 2019

Question 62:

Electrons with spin in the 𝑧-direction (𝑧̂) are passed through a Stern-Gerlach (SG) set up with the magnetic field at 𝜃 = 60° from 𝑧̂. The fraction of electrons that will emerge with their spin parallel to the magnetic field in the SG set up (rounded off to two decimal places) is __________________

[σ_{x} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{y} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), σ_{z} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})]

0.72 to 0.75

0.73 to 0.75

0.74 to 0.75

0.75 to 0.75

Category: GATE Physics 2019

Question 63:

The Hamiltonian of a system is

H = (\begin{matrix} 1 & ε \\ ε & - 1 \end{matrix})

with 𝜀 ≪ 1. The fourth order contribution to the ground state energy of

H i s γ ε^{4},

. The value of 𝛾 (rounded off to three decimal places) is _________

0.120 to 0.125

0.125 to 0.125

0.125 to 0.130

0.130 to 0.130

Category: GATE Physics 2019

Question 64:

Two events, one on the earth and the other one on the Sun, occur simultaneously in the earth’s frame. The time difference between the two events as seen by an observer in a spaceship moving with velocity 0.5c in the earth’s frame along the line joining the earth to the Sun is Δ𝑡, where c is the speed of light. Given that light travels from the Sun to the earth in 8.3 minutes in the earth’s frame, the value of |Δ𝑡| in minutes (rounded off to two decimal places) is ________________ (Take the earth’s frame to be inertial and neglect the relative motion between the earth and the sun)

4.77 to 4.80

4.77 to 4.81

4.77 to 4.82

4.77 to 4.83

Category: GATE Physics 2019

Question 65:

In a certain two-dimensional lattice, the energy dispersion of the electrons is

$ε (\vec{k}) = - 2 t [\cos k_{x} α + 2 \cos \frac{1}{2} k_{x} α \cos \frac{\sqrt{3}}{2} k_{y} α]$

where

\vec{k} = (k_{x}, k_{y})

denotes the wave vector, 𝑎 is the lattice constant and 𝑡 is a constant in units of eV. In this lattice the effective mass tensor

m_{i j}

of electrons calculated at the center of the Brillouin zone has the form

m_{i j} = \frac{ℏ^{2}}{t α^{2}} (\begin{matrix} α & 0 \\ 0 & α \end{matrix}) .

. The value of 𝛼 (rounded off to three decimal places) is ___________

0.330 to 0.333

0.331 to 0.333

0.332 to 0.333

0.333 to 0.333

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Answered & Marked for Review (will be considered for evaluation)

Question 1:

The fishermen, _______ the flood victims owed their lives, were rewarded by the government.

Correct Answer : to whom

Question 2:

Correct Answer : 4

Question 3:

The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is ______.

Correct Answer : 33.1

Question 4:

Correct Answer : 7

Question 5:

Until Iran came along, India had never been ________________ in kabaddi.

Correct Answer : defeated

Question 6:

A	Whenever the Reserve Bank of India reduces the repo rate, the interest rates on small saving schemes are also reduced
B	Interest rates on small saving schemes are always maintained on par with fixed deposit interest rates
C	The government sometimes takes into consideration the demands of banking institutions before reducing the interest rates on small saving schemes
D	A reduction in interest rates on small saving schemes follow only after a reduction in repo rate by the Reserve Bank of India

Correct Answer : C

Question 7:

Correct Answer : 10.50

Question 8:

Correct Answer : baaj, institution

Question 9:

Correct Answer : 10

Question 10:

Correct Answer : the number of heads he could gather

Question 11:

The relative magnetic permeability of a type-I superconductor is

A	0	B	-1
C	2π	D	$\frac{1}{4 π}$

Correct Answer : A

Question 12:

Considering baryon number and lepton number conservation laws, which of the following processes is/are allowed?

(i) ρ \to π^{0} + e^{+} + v_{e} (ii) e^{+} + v_{e} \to μ^{+} + v_{μ}

Correct Answer : only (ii)

Question 13:

For the following circuit, what is the magnitude of

V_{o u t} i f V_{i n} = 1.5 V

Correct Answer : 15 V

Question 14:

For the differential equation

\frac{d^{2} y}{d x^{2}} - n (n + 1) \frac{y}{x^{2}} = 0,

, where 𝑛 is a constant, the product of its two independent solutions is

A	$\frac{1}{x}$	B	x
C	$x^{n}$	D	$\frac{1}{x^{n + 1}}$

Correct Answer : B

Question 15:

Consider a one-dimensional gas of 𝑁 non-interacting particles of mass 𝑚 with the Hamiltonian for a single particle given by,

H = \frac{ρ^{2}}{2 m} + \frac{1}{2} m ω^{2} (x^{2} + 2 x)

The high temperature specific heat in units of 𝑅 =

N k_{B}

(

k_{B}

is the Boltzmann constant) is

Correct Answer : 1

Question 16:

An electric field

\vec{E} = E_{0} \hat{z}

A	⟨0,0\|𝐻′\|0,0⟩	B	⟨0,0\|𝐻′\|1,1⟩
C	⟨0,0\|𝐻′\|1,0⟩	D	⟨0,0\|𝐻′\|1, −1⟩

Correct Answer : C

Question 17:

A large number 𝑁 of ideal bosons, each of mass m, are trapped in a three-dimensional potential

V (r) = \frac{m ω^{2} r^{2}}{2} .

. The bosonic system is kept at temperature 𝑇 which is much lower than the Bose-Einstein condensation temperature

T_{C}

. The chemical potential (𝜇) satisfies

A	$μ \leq \frac{3}{2} ℏ ω$	B	$2 ℏ ω > μ > \frac{3}{2} ℏ ω$
C	$3 ℏ ω > μ > 2 ℏ ω$	D	$μ = 3 ℏ ω$

Correct Answer : A

Question 18:

During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix

(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 1 \\ - 1 & 0 & 0 \end{matrix})

, the unit vector along the axis of rotation is

A	$\frac{1}{3} (2 \hat{i} - \hat{j} + 2 \hat{k})$	B	$\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} - \hat{k})$
C	$\frac{1}{\sqrt{3}} (\hat{i} - \hat{j} - \hat{k})$	D	$\frac{1}{3} (2 \hat{i} + 2 \hat{j} - \hat{k})$

Correct Answer : B

Question 19:

For a spin

\frac{1}{2}

particle, let | ↑⟩ and | ↓⟩ denote its spin up and spin down states, respectively. If |𝑎⟩ =

\frac{1}{\sqrt{2}}

( | ↑⟩ | ↓⟩ + | ↓⟩ | ↑⟩) and |𝑏⟩ =

\frac{1}{\sqrt{2}}

( | ↑⟩ | ↓⟩ − | ↓⟩ | ↑⟩) are composite states of two such particles, which of the following statements is true for their total spin S ?

A	S=1 for \|𝑎⟩, and \|𝑏⟩ is not an eigenstate of the operator ${\hat{S}}^{2}$
B	\|𝑎⟩ is not an eigenstate of the operator ${\hat{S}}^{2}$ , and S=0 for \|𝑏⟩
C	S=0 for \|𝑎⟩, and S=1 for \|𝑏⟩
D	S=1 for \|𝑎⟩, and S=0 for \|𝑏⟩

Correct Answer : D

Question 20:

Consider a transformation from one set of generalized coordinate and momentum (𝑞, 𝑝) to another set (𝑄, 𝑃) denoted by,

Q = p q^{s}; P = q^{r}

where 𝑠 and 𝑟 are constants. The transformation is canonical if

A	𝑠 = 0 and 𝑟 = 1	B	𝑠 = 2 and 𝑟 = −1
C	𝑠 = 0 and 𝑟 = −1	D	𝑠 = 2 and 𝑟 = 1

Correct Answer : B

Question 21:

In order to estimate the specific heat of phonons, the appropriate method to apply would be

A	Einstein model for acoustic phonons and Debye model for optical phonons
B	Einstein model for optical phonons and Debye model for acoustic phonons
C	Einstein model for both optical and acoustic phonons
D	Debye model for both optical and acoustic phonons

Correct Answer : B

Question 22:

The pole of the function 𝑓(𝑧) = cot 𝑧 at 𝑧 = 0 is

Correct Answer : a simple pole

Question 23:

A massive particle X in free space decays spontaneously into two photons. Which of the following statements is true for X?

A	X is charged	B	Spin of X must be greater than or equal to 2
C	X is a boson	D	X must be a baryon

Correct Answer : C

Question 24:

The electric field of an electromagnetic wave is given by

\vec{E} = 3 \sin (k z - ω t) \hat{x} + 4 \cos (k z - ω t) \hat{y} .

The wave is

A	linearly polarized at an angle $\tan^{- 1} (\frac{4}{3})$ from the x-axis
B	linearly polarized at an angle $\tan^{- 1} (\frac{3}{4})$ from the x-axis
C	elliptically polarized in clockwise direction when seen travelling towards the observer
D	elliptically polarized in counter-clockwise direction when seen travelling towards the observer

Correct Answer : D

Question 25:

The nuclear spin and parity of

C_{20}^{40} a

in its ground state is

A	$0^{+}$	B	$0^{-}$
C	$1^{+}$	D	$1^{-}$

Correct Answer : A

Question 26:

An infinitely long thin cylindrical shell has its axis coinciding with the z-axis. It carries a surface charge density

σ_{0}

cos𝜙, where 𝜙 is the polar angle and

σ_{0}

is a constant. The magnitude of the electric field inside the cylinder is

A	0	B	$\frac{σ_{0}}{2 \in_{0}}$
C	$\frac{σ_{0}}{3 \in_{0}}$	D	$\frac{σ_{0}}{4 \in_{0}}$

Correct Answer : B

Question 27:

Consider a three-dimensional crystal of 𝑁 inert gas atoms. The total energy is given by

U (R) = [p {(\frac{σ}{R})}^{12} - q {(\frac{σ}{R})}^{6}],

Correct Answer : 1.07 to 1.11

Question 28:

The energy-wavevector (𝐸-𝑘) dispersion relation for a particle in two dimensions is 𝐸 = 𝐶𝑘, where 𝐶 is a constant. If its density of states 𝐷(𝐸) is proportional to

E^{p}

then the value of 𝑝 is ______________

Correct Answer : 1 to 1

Question 29:

A circular loop made of a thin wire has radius 2 cm and resistance 2 Ω. It is placed perpendicular to a uniform magnetic field of magnitude

|{\vec{B}}_{0}|

= 0.01

Tesla. At time 𝑡 = 0 the field starts decaying as

\vec{B} = {\vec{B}}_{0} e^{- t / t_{0}}, w h e r e t_{0} = 1 s .

The total charge that passes through a cross section of the wire during the decay is Q. The value of Q in 𝜇𝐶 (rounded off to two decimal places) is________________

Correct Answer : 6.26 to 6.30

Question 30:

The electric field of an electromagnetic wave in vacuum is given by

\vec{E} = E_{0} \cos (3 y + 4 z - 1.5 x 10^{9} t) \hat{x} .

The wave is reflected from the 𝑧 = 0 surface. If the pressure exerted on the surface is

α \in_{0} E_{0}^{2},

the value of 𝛼 (rounded off to one decimal place) is ___________

Correct Answer : 0.8 to 0.8

Question 31:

The Hamiltonian for a quantum harmonic oscillator of mass 𝑚 in three dimensions is

H = \frac{p^{2}}{2 m} + \frac{1}{2} m ω^{2} r^{2}

where 𝜔 is the angular frequency. The expectation value of

r^{2}

in the first excited state of the oscillator in units of

\frac{ℏ}{m ω}

(rounded off to one decimal place) is __________

Correct Answer : 2.5 to 2.5

Question 32:

The Hamiltonian for a particle of mass 𝑚 is 𝐻

= \frac{p^{2}}{2 m} + k q t

where 𝑞 and 𝑝 are the generalized coordinate and momentum, respectively, 𝑡 is time and 𝑘 is a constant. For the initial codition,

q = 0 a n d p = 0 a t t = 0, q (t)

\propto t^{α}

. The value of α is _______

Correct Answer : 3 to 3

Question 33:

At temperature T Kelvin (K), the value of the Fermi function at an energy 0.5 eV above the Fermi energy is 0.01. Then T, to the nearest integer, is_____________

(k_{B} = 8, 62 x 10^{- 5} e V / K)

Correct Answer : 1260 to 1266

Question 34:

Let

| {��}_{1} 〉 = (\begin{matrix} 1 \\ 0 \end{matrix}), | {��}_{2} 〉 = (\begin{matrix} 0 \\ 1 \end{matrix})

represent two possible states of a two-level quantum system. The state obtained by the incoherent superposition of

| {��}_{1} ⟩ a n d | {��}_{2}

is given by a density matrix that is defined as

�� \equiv c_{1} | {��}_{1} 〉 〈 {��}_{1} | + c_{2} | {��}_{2} 〉 〈 {��}_{2} | . I f c_{1} = 0.4 a n d {��}_{2} = 0.6,

= 0.6, the matrix element

p_{22}

(rounded off to one decimal place) is _________

Correct Answer : 0.6 to 0.6

Question 35:

Correct Answer : 0.244 to 0.248

Question 36:

Consider the following Boolean expression:

(A + B) [A (B + C)] + A (B + \bar{C})

It can be represented by a single three-input logic gate. Identify the gate.

Correct Answer : NAND

Question 37:

The value of the integral

\int_{- \infty}^{\infty} \frac{\cos (k x)}{x^{2} + α^{2}} d x, w h e r e k > 0 a n d α > 0, i s

A	$\frac{π}{α} e^{- k α}$	B	$\frac{2 π}{α} e^{- k α}$
C	$\frac{π}{2 α} e^{- k α}$	D	$\frac{3 π}{2 α} e^{- k α}$

Correct Answer : A

Question 38:

The wave function 𝜓(𝑥) of a particle is as shown below

Here 𝐾 is a constant, and 𝑎 > 𝑑. The position uncertainty 〈Δ𝑥〉 of the particle is

A	$\sqrt{\frac{α^{2} + 3 d^{2}}{12}}$	B	$\sqrt{\frac{3 α^{2} + d^{2}}{12}}$
C	$\sqrt{\frac{d^{2}}{6}}$	D	$\sqrt{\frac{d^{2}}{24}}$

Correct Answer : B

Question 39:

A	$Q R^{2} ω$	B	$\frac{1}{2} Q R^{2} ω$
C	$\frac{1}{4} Q R^{2} ω$	D	$\frac{1}{8} Q R^{2} ω$

Correct Answer : C

Question 40:

Consider the motion of a particle along the x-axis in a potential 𝑉(𝑥) = 𝐹|𝑥|. Its ground state energy

E_{0}

is estimated using the uncertainty principle. Then

E_{0}

is proportional to

A	$F^{\frac{1}{3}}$	B	$F^{\frac{1}{2}}$
C	$F^{\frac{2}{5}}$	D	$F^{\frac{2}{3}}$

Correct Answer : D

Question 41:

A 3-bit analog-to-digital converter is designed to digitize analog signals ranging from 0 V to 10 V. For this converter, the binary output corresponding to an input of 6 V is

Correct Answer : 100

Question 42:

The Hamiltonian operator for a two-level quantum system is 𝐻 =

(\begin{matrix} E_{1} & 0 \\ 0 & E_{2} \end{matrix})

. If the state of the system at 𝑡 = 0 is given by |𝜓 (0)〉 =

\frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix})

then

| ⟨ ψ (0) | ψ (𝑡) ⟩ |^{2}

at a later time 𝑡 is

A	$\frac{1}{2} (1 + e^{- (E_{1} - E_{2}) t / ℏ})$	B	$\frac{1}{2} (1 - e^{- (E_{1} - E_{2}) t / ℏ})$
C	$\frac{1}{2} (1 + \cos [(E_{1} - E_{2}) t / ℏ])$	D	$\frac{1}{2} (1 - \cos [(E_{1} - E_{2}) t / ℏ])$

Correct Answer : C

Question 43:

A particle of mass m moves in a lattice along the x-axis in a periodic potential 𝑉(𝑥) = 𝑉(𝑥 + 𝑑) with periodicity d. The corresponding Brillouin zone extends from

- k_{0} t o k_{0}

with these two k-points being equivalent. If a weak force F in the x-direction is applied to the particle, it starts a periodic motion with time period T. Using the equation of motion

F = \frac{d p c r y s t a l}{d t}

for a particle moving in a band, where 𝑝crystal is the crystal momentum of the particle, the period T is found to be (ℎ is Planck constant)

A	$\sqrt{\frac{2 m d}{F}}$	B	$2 \sqrt{\frac{2 m d}{F}}$
C	$\frac{2 h}{F d}$	D	$\frac{h}{F d}$

Correct Answer : D

Question 44:

Consider a potential barrier 𝑉(𝑥) of the form:

where

V_{0}

is a constant. For particles of energy 𝐸 <

V_{0}

incident on this barrier from the left, which of the following schematic diagrams best represents the probability density

| ψ (𝑥) |^{2}

as a function of 𝑥.

Correct Answer : A

Question 45:

The spin-orbit interaction term of an electron moving in a central field is written as

f (r) \vec{l} \cdot \vec{s},

, where r is the radial distance of the electron from the origin. If an electron moves inside a uniformly charged sphere, then

A	$f (r) = c o n s \tan t$	B	$f (r) \propto r^{- 1}$
C	$f (r) \propto r^{- 2}$	D	$f (r) \propto r^{- 3}$

Correct Answer : A

Question 46:

For the following circuit, the correct logic values for the entries

X_{2} a n d Y_{2}

in the truth table are

Correct Answer : 1 and 0

Question 47:

In a set of 𝑁 successive polarizers, the

m^{t h}

polarizer makes an angle

(\frac{m π}{2 N})

with the vertical. A vertically polarized light beam of intensity 𝐼0 is incident on two such sets with

N = N_{1} a n d N = N_{2}

, where

N_{2} > N_{1}

. Let the intensity of light beams coming out be 𝐼(

N_{1}

) and 𝐼(

N_{2}

) , respectively. Which of the following statements is correct about the two outgoing beams?

A	$I (N_{2}) > I (N_{1})$ ; the polarization in each case is vertical
B	$I (N_{2}) < I (N_{1})$ ; the polarization in each case is vertical
C	$I (N_{2}) > I (N_{1})$ ; the polarization in each case is horizontal
D	$I (N_{2}) < I (N_{1})$ ; the polarization in each case is horizontal

Correct Answer : C

Question 48:

A ball bouncing off a rigid floor is described by the potential energy function

V (x) = m g x f o r x > 0 = \infty f o r x \leq 0

Which of the following schematic diagrams best represents the phase space plot of the ball?

Correct Answer : B

Question 49:

\vec{B}

inside the superconductor is zero so that the field just outside the superconductor is parallel to its surface. The magnetic field due to this configuration at a point (𝑥, 𝑦, 𝑧 > 0) is

A	$(\frac{μ_{0} I}{2 π}) \frac{- (z - d) \hat{j} + y \hat{k}}{[y^{2} + {(z - d)}^{2}]}$
B	$(\frac{μ_{0} I}{2 π}) [\frac{- (z - d) \hat{j} + y \hat{k}}{y^{2} + {(z - d)}^{2}} + \frac{(z + d) \hat{j} - y \hat{k}}{y^{2} + {(z + d)}^{2}}]$
C	$(\frac{μ_{0} I}{2 π}) [\frac{- (z - d) \hat{j} + y \hat{k}}{y^{2} + {(z - d)}^{2}} - \frac{(z + d) \hat{j} - y \hat{k}}{y^{2} + {(z + d)}^{2}}]$
D	$(\frac{μ_{0} I}{2 π}) [\frac{y \hat{j} (z - d) \hat{k}}{y^{2} + {(z - d)}^{2}} + \frac{y \hat{j} - (z + d) \hat{k}}{y^{2} + {(z + d)}^{2}}]$

Correct Answer : B

Question 50:

The vector potential inside a long solenoid, with n turns per unit length and carrying current 𝐼, written in cylindrical coordinates is

\vec{A}

(𝑠,𝜙, 𝑧) =

\frac{μ_{0} n I}{2}

𝑠 𝜙̂. If the term

\frac{μ_{0} n I}{2}

s(𝛼 cos𝜙 𝜙̂ + 𝛽 sin𝜙 𝑠̂), where 𝛼 ≠ 0 , 𝛽 ≠ 0, is added to

\vec{A}

(𝑠,𝜙, 𝑧), the magnetic field remains the same if

A	$α = β$	B	$α = - β$
C	$α = 2 β$	D	$α = \frac{β}{2}$

[\begin{matrix} U s e f u l F o r m u l a e : \vec{\nabla} = \frac{�� t}{�� s} \hat{s} + \frac{1}{s} \frac{�� t}{�� ϕ} \hat{ϕ} + \frac{�� t}{�� z} \hat{z}; \\ \vec{\nabla} x \vec{v} = (\frac{1}{s} \frac{�� v_{z}}{�� ϕ} - \frac{�� v_{ϕ}}{�� z}) \hat{s} + (\frac{�� v_{s}}{�� z} - \frac{�� v_{z}}{�� s}) \hat{ϕ} + \frac{1}{s} (\frac{�� (s v_{ϕ}}{�� s} - \frac{�� v_{s}}{�� ϕ}) \hat{z} \end{matrix}]

Correct Answer : D

Question 51:

Low energy collision (s-wave scattering) of pion

(π^{+})

with deuteron (d) results in the production of two protons (

π^{+} +

d → p + p). The relative orbital angular momentum (in units of ℏ) of the resulting two-proton system for this reaction is

Correct Answer : 1

Question 52:

Consider the Hamiltonian

H (q, p) = \frac{α p^{2} q^{4}}{2} + \frac{β}{q^{2}},

, where 𝛼 and 𝛽 are parameters with appropriate dimensions, and 𝑞 and 𝑝 are the generalized coordinate and momentum, respectively. The corresponding Lagrangian 𝐿(𝑞, 𝑞̇) is

A	$\frac{1}{2 α} \frac{q^{2}}{q^{4}} - \frac{β}{q^{2}}$	B	$\frac{1}{α} \frac{q^{2}}{q^{4}} - \frac{β}{q^{2}}$
C	$\frac{1}{α} \frac{q^{2}}{q^{4}} + \frac{β}{q^{2}}$	D	$- \frac{1}{2 α} \frac{q^{2}}{q^{4}} + \frac{β}{q^{2}}$

Correct Answer : A

Question 53:

For a given load resistance

R_{L}

= 4.7 ohm, the power transfer efficiencies

(η = \frac{P_{l o a d}}{P_{t o t a l}})

of a dc voltage source and a dc current source with internal resistances

R_{1} a n d R_{2}

, respectively, are equal. The product

R_{1} R_{2}

in units of

o h m^{2}

(rounded off to one decimal place) is

Correct Answer : 22.0 to 22.2

Question 54:

The ground state electronic configuration of the rare-earth ion

(N d^{3 +}) i s [P d] 4 f^{3} 5 s^{2} 5 p^{6} .

Assuming LS coupling, the Land𝑒́ 𝑔-factor of this ion is

\frac{8}{11}

. The effective magnetic moment in units of Bohr magneton

μ_{B}

(rounded off to two decimal places) is

Correct Answer : 3.61 to 3.63

Question 55:

A projectile of mass 1 kg is launched at an angle of 30° from the horizontal direction at 𝑡 = 0 and takes time 𝑇 before hitting the ground. If its initial speed is 10

m s^{- 1}

, the value of the action integral for the entire flight in the units of

K g m^{2} s^{- 1}

(rounded off to one decimal place) is _____________ [ Take g =10

m s^{- 2}

]

Correct Answer : 33.2 to 33.4

Question 56:

Let 𝜃 be a variable in the range − 𝜋 ≤ 𝜃 < 𝜋. Now consider a function

ψ (θ) = 1 f o r - π / 2 \leq θ < π / 2 = 0 o t h e r w i s e .

If its Fourier-series is written as

ψ (θ) = \sum_{m = - \infty}^{\infty} C_{m} e^{- i m θ}

, then the value of

{|C_{3}|}^{2}

(rounded off to three decimal places) is _________________

Correct Answer : 0.010 to 0.013

Question 57:

Two spaceships A and B, each of the same rest length 𝐿, are moving in the same direction with speeds

\frac{4 c}{5}

and

\frac{3 c}{5},

respectively, where 𝑐 is the speed of light. As measured by B, the time taken by A to completely overtake B [see figure below] in units of 𝐿/𝑐 (to the nearest integer) is _______________

Correct Answer : 5 to 5

Question 58:

Correct Answer : 40 to 40

Question 59:

Correct Answer : 1.02 to 1.04

Question 60:

Consider a system of three charges as shown in the figure below:

For 𝑟 = 10 m; 𝜃 = 60 degrees; 𝑞 =

10^{- 6}

Coulomb, and 𝑑 =

10^{- 3}

m, the electric dipole potential in volts (rounded off to three decimal places) at a point (𝑟, 𝜃) is __________________

[U s e : \frac{1}{4 π \in_{0}} = 9 x 10^{9} \frac{N m^{2}}{C^{2}}]

Correct Answer : 0.044 to 0.046

Question 61:

K_{B}

ln 𝜆. The value of 𝜆 is________________

Correct Answer : 12 to 12

Question 62:

[σ_{x} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{y} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), σ_{z} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})]

Correct Answer : 0.75 to 0.75

Question 63:

The Hamiltonian of a system is

H = (\begin{matrix} 1 & ε \\ ε & - 1 \end{matrix})

with 𝜀 ≪ 1. The fourth order contribution to the ground state energy of

H i s γ ε^{4},

. The value of 𝛾 (rounded off to three decimal places) is _________

Correct Answer : 0.125 to 0.125

Question 64:

Correct Answer : 4.77 to 4.81

Question 65:

In a certain two-dimensional lattice, the energy dispersion of the electrons is

$ε (\vec{k}) = - 2 t [\cos k_{x} α + 2 \cos \frac{1}{2} k_{x} α \cos \frac{\sqrt{3}}{2} k_{y} α]$

where

\vec{k} = (k_{x}, k_{y})

denotes the wave vector, 𝑎 is the lattice constant and 𝑡 is a constant in units of eV. In this lattice the effective mass tensor

m_{i j}

of electrons calculated at the center of the Brillouin zone has the form

m_{i j} = \frac{ℏ^{2}}{t α^{2}} (\begin{matrix} α & 0 \\ 0 & α \end{matrix}) .

. The value of 𝛼 (rounded off to three decimal places) is ___________

Correct Answer : 0.333 to 0.333

Back
Scrore Card
Total Question		Total Attempted
Correct Answers		Incorrect Answers
Score

A	⟨0,0\|𝐻′\|0,0⟩	B	⟨0,0\|𝐻′\|1,1⟩
C	⟨0,0\|𝐻′\|1,0⟩	D	⟨0,0\|𝐻′\|1, −1⟩

A	S=1 for \|𝑎⟩, and \|𝑏⟩ is not an eigenstate of the operator ${\hat{S}}^{2}$
B	\|𝑎⟩ is not an eigenstate of the operator ${\hat{S}}^{2}$ , and S=0 for \|𝑏⟩
C	S=0 for \|𝑎⟩, and S=1 for \|𝑏⟩
D	S=1 for \|𝑎⟩, and S=0 for \|𝑏⟩

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