List of top Questions asked in GATE PH

“Why do they pull down and do away with crooked streets, I wonder, which are my delight, and hurt no man living? Every day the wealthier nations are pulling down one or another in their capitals and their great towns: they do not know why they do it; neither do I. It ought to be enough, surely, to drive the great broad ways which commerce needs and which are the life-channels of a modern city, without destroying all history and all the humanity in between: the islands of the past.”
(From Hilaire Belloc’s “The Crooked Streets”)
Based only on the information provided in the above passage, which one of the following statements is true?

As the police officer was found guilty of embezzlement, he was _________ dismissed from the service in accordance with the Service Rules. Select the most appropriate option to complete the above sentence.

A wheel of mass \( 4M \) and radius \( R \) is made of a thin uniform distribution of mass \( 3M \) at the rim and a point mass \( M \) at the center. The spokes of the wheel are massless. The center of mass of the wheel is connected to a horizontal massless rod of length \( 2R \), with one end fixed at \( O \), as shown in the figure. The wheel rolls without slipping on horizontal ground with angular speed \( \Omega \). If \( \vec{L} \) is the total angular momentum of the wheel about \( O \), then the magnitude \( \left| \frac{d\vec{L}}{dt} \right| = N(MR^2 \Omega^2) \). The value of \( N \) (in integer) is:


 

An electron with mass \( m \) and charge \( q \) is in the spin up state \[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] at time \( t = 0 \). A constant magnetic field is applied along the \( y \)-axis, \( \mathbf{B} = B_0 \hat{j} \), where \( B_0 \) is a constant. The Hamiltonian of the system is \[ H = -\hbar \omega \sigma_y, \quad \text{where} \quad \omega = \frac{q B_0}{2m} < 0 \quad \text{and} \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}. \]
The minimum time after which the electron will be in the spin down state along the \( x \)-axis, i.e., \[ \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}, \] is:

The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop. The opamp runs from \( \pm 15 \, {V} \) supplies. If a \( +1 \, {V} \) signal is applied at the input, the output voltage (rounded off to one decimal place) is:


 

A system of five identical, non-interacting particles with mass \( m \) and spin \( \frac{3}{2} \) is confined to a one-dimensional potential well of length \( L \). If the lowest energy of the system is \[ E_{{min}} = \frac{N \pi^2 \hbar^2}{2m L^2}, \] the value of \( N \) (in integer) is:
The Hamiltonian for a one-dimensional system with mass \( m \), position \( q \), and momentum \( p \) is: \[ H(p, q) = \frac{p^2}{2m} + q^2 A(q) \] where \( A(q) \) is a real function of \( q \). If \[ m \frac{d^2 q}{dt^2} = -5q A(q), \] then \[ \frac{d A(q)}{d q} = n \frac{A(q)}{q}. \] The value of \( n \) (in integer) is:

In the transistor circuit shown in the figure, \( V_{BE} = 0.7 \, {V} \) and \( \beta_{DC} = 400 \). The value of the base current in \( \mu A \) (rounded off to one decimal place) is:


 

Cyclotrons are used to accelerate ions like deuterons (\( d \)) and \( \alpha \) particles. Keeping the magnetic field the same for both, \( d \) and \( \alpha \) are extracted with energies 10 MeV and 20 MeV with extraction radii \( r_d \) and \( r_\alpha \), respectively. Taking the masses \( M_d = 2000 \, {MeV}/c^2 \) and \( M_\alpha = 4000 \, {MeV}/c^2 \), the value of \( \frac{r_\alpha}{r_d} \) (in integer) is:
A linear magnetic material in the form of a cylinder of radius \( R \) and length \( L \) is placed with its axis parallel to the \( z \)-axis. The cylinder has uniform magnetization \( M \hat{k} \). Which of the following option(s) is/are correct?
A point charge \( q \) is placed at the origin, inside a linear dielectric medium of infinite extent, having relative permittivity \( \epsilon_r \). Which of the following option(s) is/are correct?
A neutral conducting sphere of radius \( R \) is placed in a uniform electric field of magnitude \( E_0 \), that points along the \( z \)-axis. The electrostatic potential at any point \( r \) outside the sphere is given by: \[ V(r, \theta) = V_0 - E_0 r \left(1 - \frac{R^3}{r^3} \right) \cos \theta \] where \( V_0 \) is the constant potential of the sphere. Which of the following option(s) is/are correct?
Consider the integral \[ I = \frac{1}{2 \pi i} \oint \frac{1}{(z^4 - 1)(z - \frac{a}{b})(z - \frac{b}{a})} \, dz, \] where \( z \) is a complex variable and \( a, b \) 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?
The energy of a free, relativistic particle of rest mass \( m \) moving along the \( x \)-axis in one dimension, is denoted by \( T \). When moving in a given potential \( V(x) \), its Hamiltonian is \( H = T + V(x) \). In the presence of this potential, its speed is \( v \), conjugate momentum \( p \), and the Lagrangian \( L \). Then, which of the following option(s) is/are correct?

The Lagrangian of a particle of mass \( m \) and charge \( q \) moving in a uniform magnetic field of magnitude \( 2B \) that points in the \( z \)-direction, is given by: \[ L = \frac{m}{2} v^2 + qB(x v_y - y v_x) \] where \( v_x, v_y, v_z \) are the components of its velocity \( v \). If \( p_x, p_y, p_z \) denote the conjugate momenta in the \( x, y, z \)-directions and \( H \) is the Hamiltonian, which of the following option(s) is/are correct?

In coordinates \( (t, x) \), a contravariant second rank tensor \( A \) has non-zero diagonal components \( A^{tt} = P \) and \( A^{xx} = Q \), with all other components vanishing, and \( P, Q \) being real constants. Here, \( t \) is time and \( x \) is space coordinate. Consider a Lorentz transformation \( (t, x) \to (t', x') \) to another frame that moves with relative speed \( v \) in the \( +x \) direction, so that \( A \to A' \). If \( A'^{tt} \) and \( A'^{xx} \) are the diagonal components of \( A' \), then setting the speed of light \( c = 1 \), and with \( \gamma = \frac{1}{\sqrt{1 - v^2}} \), which of the following option(s) is/are correct?

A point charge \( q \) is placed at a distance \( d \) above an infinite, grounded conducting plate placed on the \( xy \)-plane at \( z = 0 \).
The electrostatic potential in the \( z > 0 \) region is given by \( \phi = \phi_1 + \phi_2 \), where:

\( \phi_1 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{q}{\sqrt{x^2 + y^2 + (z - d)^2}} \)
\( \phi_2 = - \frac{1}{4 \pi \epsilon_0} \cdot \frac{q}{\sqrt{x^2 + y^2 + (z + d)^2}} \)

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

Consider two hypothetical nuclei \( X_1 \) and \( X_2 \) undergoing \( \beta \) decay, resulting in nuclei \( Y_1 \) and \( Y_2 \), respectively. The decay scheme and the corresponding \( J^P \) values of the nuclei are given in the figure. Which of the following option(s) is/are correct? (\( J \) is the total angular momentum and \( P \) is parity)


 

Which of the following option(s) is/are correct for a Type I superconductor?
A system of three non-identical spin-\(\frac{1}{2}\) particles has the Hamiltonian
\[ H = \frac{A \hbar^2}{2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3, \] where \( \vec{S}_1, \vec{S}_2, \vec{S}_3 \) are the spin operators of particles labelled 1, 2, and 3 respectively, and \( A \) is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is:

A two-level quantum system has energy eigenvalues \( E_1 \) and \( E_2 \). A perturbing potential \( H' = \lambda \Delta \sigma_x \) is introduced, where \( \Delta \) is a constant having dimensions of energy, \( \lambda \) is a small dimensionless parameter, and \( \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \).

The magnitudes of the first and the second order corrections to \( E_1 \) due to \( H' \), respectively, are:

Let \( |m\rangle \) and \( |n\rangle \) denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are \( \hat{X} \) and \( \hat{P} \), respectively. The matrix element \( \langle m | \hat{P}\hat{X} | n \rangle \) is non-zero when:

Consider a two-level system with energy states \( +\epsilon \) and \( -\epsilon \). The number of particles at \( +\epsilon \) level is \( N+ \) and the number of particles at \( -\epsilon \) level is \( N- \). The total energy of the system is \( E \) and the total number of particles is \( N = N+ + N- \). In the thermodynamic limit, the inverse of the absolute temperature of the system is: 
(Given: \( \ln N! \approx N \ln N - N \))

An \( \alpha \) particle is scattered from an Au target at rest as shown in the figure. \( D_1 \) and \( D_2 \) are the detectors to detect the scattered \( \alpha \) particle at an angle \( \theta \) and along the beam direction, respectively, as shown. The signals from \( D_1 \) and \( D_2 \) are converted to logic signals and fed to logic gates. When a particle is detected, the signal is 1 and is 0 otherwise. Which one of the following circuits detects the particle scattered at the angle \( \theta \) only?


 

A thin circular ring of radius \( R \) lies in the \( xy \)-plane with its centre coinciding with the origin. The ring carries a uniform line charge density \( \lambda \). The quadrupole contribution to the electrostatic potential at the point \( (0, 0, d) \), where \( d \gg R \), is: