List of top Questions asked in GATE PH

Consider a two-dimensional Cartesian coordinate system in which a rank 2 contravariant tensor is represented by the matrix \[ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \] The coordinate system is rotated anticlockwise by an acute angle $\theta$ with the origin fixed. Which one of the following matrices represents the tensor in the new coordinate system?

The Hall experiment is carried out with a non-magnetic semiconductor. The current $I$ is along the X-axis and the magnetic field $B$ is along the Z-axis. Which one of the following is the CORRECT representation of the variation of the magnitude of the Hall resistivity $\rho_{xy}$ as a function of the magnetic field?

H is the Hamiltonian, $\vec{L}$ the orbital angular momentum, and $L_z$ is the z-component of $\vec{L}$. The 1S state of the hydrogen atom in the non-relativistic formalism is an eigenfunction of which one of the following sets of operators?
Choose the most appropriate matching of the items in Column 1 with those in Column 2. \[ \begin{array}{|c|c|} \hline \textbf{Column 1} & \textbf{Column 2} \\ \hline (i)\ \text{PIN diode} & Q.\ \text{Radio frequency and microwave devices} \\ (ii)\ \text{Tunnel diode} & S.\ \text{Oscillator} \\ (iii)\ \text{Zener diode} & P.\ \text{Voltage regulation} \\ (iv)\ \text{Photo diode} & R.\ \text{Optoelectronic detection} \\ \hline \end{array} \]

Which one of the following entropy (S) - temperature (T) diagrams CORRECTLY represents the Carnot cycle abcda shown in the P-V diagram? 

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ and MN form lines of symmetry? (The figure is representative)

In the given figure, PQRS is a parallelogram with $PS = 7~\text{cm}$, $PT = 4~\text{cm}$ and $PV = 5~\text{cm}$. What is the length of $RS$ in cm? (The diagram is representative.)

Two identical bodies kept at temperatures 800 K and 200 K act as the hot and the cold reservoirs of an ideal heat engine, respectively. Assume that their heat capacity ($C$) in Joules/K is independent of temperature and that they do not undergo any phase change. Then, the maximum work that can be obtained from the heat engine is $n \times C$ Joules. What is the value of $n$ (in integer)?
Two identical, non-interacting $^4He_2$ atoms are distributed among 4 different non-degenerate energy levels. The probability that they occupy different energy levels is $p$. Similarly, two $^3He_2$ atoms are distributed among 4 different non-degenerate energy levels, and the probability that they occupy different levels is $q$. What is the value of $\dfrac{p{q}$ (rounded off to one decimal place)?}
In pion nucleon scattering, the pion and nucleon can combine to form a short-lived bound state called the \(\Delta\) particle (\(\pi + N ⇒ \Delta\)). The masses of the pion, nucleon, and the \(\Delta\) particle are 140 MeV/\(c^2\), 938 MeV/\(c^2\), and 1230 MeV/\(c^2\), respectively. In the lab frame, where the nucleon is at rest, what is the minimum energy (in MeV/\(c^2\), rounded off to one decimal place) of the pion to produce the \(\Delta\) particle?
Consider the vector field \(\vec{V}\) consisting of the velocities of points on a thin horizontal disc of radius \(R = 2 \, \text{m}\), moving anticlockwise with uniform angular speed \(\omega = 2 \, \text{rad/sec}\) about an axis passing through its center. If \(V = |\vec{V}|\), then which of the following options is/are CORRECT? (In the options, \(\hat{r}\) and \(\hat{\theta}\) are unit vectors corresponding to the plane polar coordinates \(r\) and \(\theta\)). \includegraphics[width=0.75\linewidth]{image57.png}
A slow moving \(\pi^{-}\) particle is captured by a deuteron (\(d\)) and this reaction produces two neutrons (\(n\)) in the final state, i.e., \(\pi^{-} + d \to n + n\).
Neutron and deuteron have even intrinsic parities, whereas \(\pi^{-}\) has odd intrinsic parity. \(L\) and \(S\) are the orbital and spin angular momenta, respectively of the system of two neutrons. Which of the following statements regarding the final two-neutron state is/are CORRECT?
Different spectral lines of the Balmer series (transitions $n \to 2$, with $n$ being the principal quantum number) fall one at a time on a Young’s double slit apparatus. The separation between the slits is $d$ and the screen is placed at a constant distance from the slits. What factor should $d$ be multiplied by to maintain a constant fringe width for various lines, as $n$ takes different allowed values?
Consider the complex function \[ f(z) = \dfrac{z^2 \sin z}{(z - \pi)^4}. \] At $z = \pi$, which of the following options is(are) CORRECT?
For a covalently bonded solid consisting of ions of mass \(m\), the binding potential can be assumed to be given by \[ U(r) = -\varepsilon \left( \frac{r}{r_0} \right) e^{-\frac{r}{r_0}}, \] where \(\varepsilon\) and \(r_0\) are positive constants. What is the Einstein frequency of the solid in Hz?
A particle has wavefunction \[ \psi(x, y, z) = N z e^{-\alpha(x^2 + y^2 + z^2)}, \] where \(N\) is a normalization constant and \(\alpha\) is a positive constant. In this state, which one of the following options represents the eigenvalues of \(L^2\) and \(L_z\) respectively?
Some values of \(Y_\ell^m\) are: \[ Y_0^0 = \frac{1}{\sqrt{4\pi}}, \ Y_1^0 = \frac{\sqrt{3}}{\sqrt{4\pi}} \cos \theta, \ Y_1^{\pm 1} = \mp \frac{\sqrt{3}}{8\pi} \sin \theta e^{\pm i \phi} \]
Consider two real functions \[ U(x,y) = xy(x^2 - y^2), \] \[ V(x,y) = ax^4 + by^4 + cx^2y^2 + k, \] where $k$ is a real constant and $a, b, c$ are real coefficients. If $U(x, y) + i V(x, y)$ is analytic, then what is the value of $a \times b \times c$?
In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive z axis for a $2p$ electron?
For a transistor amplifier, the frequency response is such that the mid band voltage gain is 200. The cutoff frequencies are 20 Hz and 20 kHz. What is the ratio (rounded off to two decimal places) of the voltage gain at 10 Hz to that at 100 kHz?
It is given that the electronic ground state of a diatomic molecule $X_2$ has even parity and the nuclear spin of $X$ is 0. Which one of the following is the CORRECT statement with regard to the rotational Raman spectrum ($J$ is the rotational quantum number) of this molecule?
Which of the following options represent(s) linearly independent pair(s) of functions of a real variable $x$?
Which of the following is/are the CORRECT option(s) for the Joule-Thomson effect?
The deuteron is a bound state of a neutron and a proton. Which of the following statements is/are CORRECT?
The Geiger-Muller counter is a device to detect \(\alpha\), \(\beta\), and \(\gamma\) radiations. It is a cylindrical tube filled with monatomic gases like argon, and polyatomic gases such as ethyl alcohol. The inner electrode is along the axis of the cylindrical tube and the outer electrode is the tube. Which of the following statements is/are CORRECT?
A compound consists of three ions X, Y, and Z. The Z ions are arranged in an FCC arrangement. The X ions occupy \(\frac{1}{6}\) of the tetrahedral voids and the Y ions occupy \(\frac{1}{3}\) of the octahedral voids. Which one of the following is the CORRECT chemical formula of the compound?