GATE PH year:2025 PYQs & Answers — Page 6

Questions 51–60 of 65. Imported answers require independent review unless marked reviewed.

The Lagrangian of a particle of mass

and charge

moving in a uniform magnetic field of magnitude

that points in the

direction, is given by

where

are the components of its velocity

. If

denote the conjugate momenta in the

directions and

is the Hamiltonian, which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

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A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the

axis. The rod itself is moving vertically upward along the

direction against gravity with a constant speed, starting from

, and remains horizontal. The conjugate momenta are denoted by

and the Hamiltonian by

. Which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

is the total energy of the system and is conserved

is the total energy of the system and is not conserved

is not the total energy of the system, but it is conserved

is not the total energy of the system and is not conserved

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In a one-dimensional Hamiltonian system with position

and momentum

, consider the canonical transformation

, where

and

are the new position and momentum, respectively. Which of the following option(s) regarding the generating function

is/are correct?

Your answerSelect all correct options. You can choose more than one.

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The energy of a free, relativistic particle of rest mass

moving along the

axis in one dimension, is denoted by

. When moving in a given potential

, its Hamiltonian is

. In the presence of this potential, its speed is

, conjugate momentum

, and the Lagrangian

. Then, which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

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Consider the integral

where

is a complex variable and

are positive real numbers. The integral is taken over a unit circle with center at the origin. Which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

when

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A neutral conducting sphere of radius

is placed in a uniform electric field of magnitude

, that points along the

axis. The electrostatic potential at any point

outside the sphere is given by

, where

is the constant potential of the sphere. Which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

The induced surface charge density on the sphere is proportional to

The electric field at any point is curl free for

The electric field at any point is divergence free for

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A point charge

is placed at the origin, inside a linear dielectric medium of infinite extent, having relative permittivity

. Which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

The magnitude of the polarization varies as

The magnitude of the screened charge due to the dielectric medium is less than the magnitude of the point charge

for

The magnitude of the screened charge due to the dielectric medium is more than the magnitude of the point charge

for

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A linear magnetic material in the form of a cylinder of radius

and length

is placed with its axis parallel to the

axis. The cylinder has uniform magnetization

. Which of the following option(s) is/are correct?

Your answerSelect all correct options. You can choose more than one.

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

The bound volume current density is zero

The surface current density on the curved surface is non-zero

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

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Cyclotrons are used to accelerate ions like deuterons (

) and

particles. Keeping the magnetic field same for both,

and

are extracted with energies 10 MeV and 20 MeV with extraction radii

and

, respectively. Taking the masses

and

, the value of

(in integer) is ______

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In the transistor circuit shown in the figure,

and

. The value of the base current in

(rounded off to one decimal place) is ________

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GATE PH year:2025 PYQs

Question 51

Question 52

Question 53

Question 54

Question 55

Question 56

Question 57

Question 58

Question 59

Question 60