List of top Questions asked in GATE PH

A circle with center at \( (x, y) = (0.5, 0) \) and radius = 0.5 intersects with another circle with center at \( (x, y) = (1, 1) \) and radius = 1 at two points. One of the points of intersection \( (x, y) \) is:

The sum of the following infinite series is: \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \dots \]

Consider the induced nuclear fission reaction:
\[ {}^{235}_{92}\text{U} + n \rightarrow {}^{93}_{37}\text{Rb} + {}^{141}_{55}\text{Cs} + 2n \]
where neutron momenta in both initial and final states are negligible. The ratio of the kinetic energies (KE) of the daughter nuclei,
\[ \frac{\text{KE}({}^{93}_{37}\text{Rb})}{\text{KE}({}^{141}_{55}\text{Cs})} \]
is __________.

Binding energy and rest mass energy of a two-nucleon bound state are denoted by \( B \) and \( mc^2 \), respectively, where \( c \) is the speed of light. The minimum energy of a photon required to dissociate the bound state is _______.

An atom is subjected to a weak uniform magnetic field \( B \). The number of lines in its Zeeman spectrum for the transition from \( n = 2, l = 1 \) to \( n = 1, l = 0 \) is

The mean distance between the two atoms of an HD molecule is \( r \), where H and D denote hydrogen and deuterium, respectively. The mass of the hydrogen atom is \( m_H \). The energy difference between the two lowest lying rotational states of HD in multiples of \(\frac{h^2}{m_H r^2}\) is:

A particle of mass $m$ is moving in the potential 
\[ V(x) = \begin{cases} V_0 + \frac{1}{2} m \omega_{0P}^2 x^2 & \quad \text{if } x > 0 \\ \infty & \quad \text{if } x \leq 0 \end{cases} \] 
Figures P, Q, R, and S show different combinations of the values of $ \omega_0 $ and $ V_0 $.

The Lagrangian of a particle of mass \( m \) is given by \( L = \frac{1}{2}m\dot{x}^2 - \lambda x^4 \), where \( \lambda \) is a positive constant. If the particle oscillates with total energy \( E \), then the time period of oscillations \( T \) is given by:
\[ T = a \int_0^{\left(\frac{E}{\lambda}\right)^{\frac{1}{4}}} \frac{dx}{\sqrt{\frac{2}{m}(E - \lambda x^4)}} \]
The value of \( a \) is ___ (in integer).

When an unpolarized plane electromagnetic wave in a dielectric medium with refractive index \( n_1 \) is incident on a plane interface separating it from another dielectric medium with refractive index \( n_2 \) (where \( n_2 > n_1 \)), and the angle of incidence is \(\tan^{-1} \left( \frac{n_2}{n_1} \right) \), the following statement is true:

If \( F_1(Q, q) = Qq \) is the generating function of a canonical transformation from \((p, q)\) to \((P, Q)\), then which one of the following relations is correct?

Consider the Lagrangian \( L = m\dot{x}\dot{y} - mw_0^2 zxy \). If \( p_x \) and \( p_y \) denote the generalized momenta conjugate to \( x \) and \( y \), respectively, then the canonical equations of motion are _______.

Given that \( A^\alpha \) and \( B_\beta \) (\(\alpha, \beta = 1, 2, 3, \ldots, n\)) are contravariant and covariant vectors, respectively, and by convention, any repeated indices are summed over. Which of the following expressions is/are tensors?

According to the nuclear shell model, the absolute value of the difference in magnetic moments of \({^{15}_{8}O}\) and \({^{15}_{7}N}\), in the units of nuclear magneton (\(\mu\)), is \(\frac{a}{3}\). The magnitude of \(a\) is (in integer).

The X-ray diffraction pattern of a monatomic cubic crystal with rigid spherical atoms of radius 1.56 Å shows several Bragg reflections of which the reflection appearing at the lowest 2𝜃 value is from (111) plane. If the wavelength of X-ray used is 0.78 Å, the Bragg angle (in 2𝜃, rounded off to one decimal place) corresponding to this reflection and the crystal structure, respectively, are

Two point charges of charge \( +q \) each are placed a distance \( 2d \) apart. A grounded solid conducting sphere of radius \( a \) is placed midway between them. Assume \( a^2 \ll d^2 \). Which of the following statements is/are true?

If a thermodynamical system is adiabatically isolated and experiences a change in volume under an externally applied constant pressure, then the thermodynamical potential minimized at equilibrium is the

Let \( p(\bar{p}, \bar{q}, t) \) be the phase space density of an ensemble of a system. The Hamiltonian of the system is \( H(p, q) \). If \( \{A, B\} \) denotes the Poisson bracket of \( A \) and \( B \), then \( \frac{dp}{dt} = 0 \) implies

An extrinsic semiconductor shown in the figure carries a current of 2 mA along its length parallel to the \( +x \) axis.

When the majority charge carrier concentration is \( 12.5 \times 10^{13} \, \text{cm}^{-3} \) and the sample is exposed to a constant magnetic field applied along the \( +z \) direction, a Hall voltage of 20 mV is measured with the negative polarity at the \( y = 0 \) plane. Take the electric charge as \( 1.6 \times 10^{-19} \) C. The concentration of minority charge carriers is negligible. Which of the following statements is/are true?

A particle of mass \( m \) in an infinite potential well of width \( a \) is subjected to a perturbation \( V' = \frac{h^2}{40ma^2} \) as shown in the figure, where \( h \) is Planck's constant.

The first-order energy shift of the fourth energy eigenstate due to this perturbation is \( \frac{h^2}{Nma^2} \). The value of \( N \) is ____ (in integer).

Consider a three-dimensional system of non-interacting bosons with zero chemical potential. The energy of the system \( \epsilon \propto k^2 \), where \( k \) is the wave vector. The low temperature specific heat of the system at constant volume depends on the temperature as \( C_v = T^{n/2} \). The value of \( n \) is ___ (in integer).

Consider the operational amplifier circuit shown in figure.

The output voltage V。 is ____ V (in integer).

An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states \( \psi_{n i m} \):\[ \psi = \frac{1}{3} \psi_{100} + \frac{1}{\sqrt{3}} \psi_{210} + \frac{\sqrt{5}}{3} \psi_{320} \]Let \( E_1, E_2, \) and \( E_3 \) represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time \( t_1 \) and the outcome is \( E_2 \). Then total angular momentum is measured at time \( t_2 > t_1 \), and finally, energy is measured again at \( t_3 > t_2 \). The probability of finding the system in a state with energy \( E_2 \) after the final measurement is \( \frac{P}{9} \). The value of \( P \) is ______ (in integer).

The temperature \( T \) dependence of magnetic susceptibility \( \chi \) (Column I) of certain magnetic materials (Column II) are given below. Which of the following options is/are correct?

Column I	Column II
(1)	(P) Diamagnetic
(2)	(Q) Paramagnetic
(3)	(R) Ferromagnetic
(4)	(S) Antiferromagnetic

The curves \( P \) and \( Q \) schematically show the variation of X-ray intensity with wavelength at two different accelerating voltages for a given target material. In the figure, \( \lambda_1 = 0.25 \) Å, \( \lambda_2 = 0.5 \) Å, \( \lambda_3 = 1.0 \) Å, and \( \lambda_4 = 2.25 \) Å. Take Planck's constant as \( 6.6 \times 10^{-34} \) Js, the speed of light as \( 3 \times 10^8 \) m/s, and the elementary charge as \( 1.6 \times 10^{-19} \) C.

Which of the following statement is/are true?