List of practice Questions

If a function \(y(x)\) is described by the initial-value problem, \(\dfrac{d^2y}{dx^2} + 5\dfrac{dy}{dx} + 6y = 0\), with initial conditions \(y(0) = 2\) and \(\dfrac{dy}{dx}\bigg|_{x=0} = 0\), then the value of \(y\) at \(x = 1\) is ................. . (Round off to 2 decimal places)

A vehicle of mass 600 kg with an engine operating at constant power \(P\) accelerates from rest on a straight horizontal road. The vehicle covers a distance of 600 m in 1 minute. Neglecting all losses, the magnitude of \(P\) is ............. kW. (Round off to 2 decimal places)

The magnetic fields in tesla in the two regions separated by the \(z = 0\) plane are given by \(\vec{B_1} = 3\hat{i} + 5\hat{j}\) and \(\vec{B_2} = 8\hat{i} + 3\hat{j} + 5\hat{k}\). The magnitude of the surface current density at the interface between the two regions is \(\alpha \times 10^6\, \text{A/m}\). Given the permeability of free space \(\mu_0 = 4\pi \times 10^{-7}\, \text{N/A}^2\), the value of \(\alpha\) is ............... . (Round off to 2 decimal places)

Consider a thin bi-convex lens of relative refractive index \(n = 1.5\). The radius of curvature of one surface of the lens is twice that of the other. The magnitude of larger radius of curvature in units of the focal length of the lens is ............. (Round off to 1 decimal place).

Water flows in a horizontal pipe in a streamlined manner at an absolute pressure of \(4 \times 10^5 \, \text{Pa}\) and speed of \(6 \, \text{m/s}\). If it exits the pipe at a pressure of \(10^5 \, \text{Pa}\), the speed of water at the exit point is .......... m/s (Round off to 1 decimal place).
(The density of water is \(1000 \, \text{kg/m}^3\))

Consider a retarder with refractive indices \(n_e = 1.551\) and \(n_o = 1.542\) along the extraordinary and ordinary axes, respectively. The thickness of this retarder for which a left circularly polarized light of wavelength \(600 \, \text{nm}\) will be converted into a right circularly polarized light is ........... µm. (Round off to 2 decimal places).

Consider two spherical perfect blackbodies with radii \(R_1\) and \(R_2\) at temperatures \(T_1 = 1000\, \text{K}\) and \(T_2 = 2000\, \text{K}\), respectively. They both emit radiation of power 1 kW. The ratio of their radii, \(R_1/R_2\), is given by ...............

In a Compton scattering experiment, the wavelength of incident X-rays is 0.500 Å. If the Compton wavelength \(\lambda_C = 0.024\, \text{Å}\), the value of the longest wavelength possible for the scattered X-ray is ............... Å. (Specify up to 3 decimal places.)

A solid with FCC crystal structure is probed using X-rays of wavelength 0.2 nm. For the crystallographic plane given by (2, 0, 0), a first-order diffraction peak is observed for a Bragg angle of \(21^\circ\). The unit cell size is ............ nm. (Round off to 2 decimal places.)

A spherical dielectric shell with inner radius \(a\) and outer radius \(b\), has polarization \(\vec{P} = \dfrac{k}{r^2}\hat{r}\), where \(k\) is a constant and \(\hat{r}\) is the unit vector along the radial direction. Which of the following statements is/are correct?

One mole of an ideal gas having specific heat ratio (\(\gamma\)) of 1.6 is mixed with one mole of another ideal gas having specific heat ratio of 1.4. If \(C_V\) and \(C_P\) are the molar specific heat capacities of the gas mixture at constant volume and pressure, respectively, which of the following is/are correct? (\(R\) denotes the universal gas constant.)

Two relativistic particles with opposite velocities collide head-on and come to rest by sticking with each other. Which of the following quantities is/are conserved in the collision?

If \(P\) and \(Q\) are Hermitian matrices, which of the following is/are true?
(A matrix \(P\) is Hermitian if \(P = P^{\dagger}\), where the elements \(p_{ij}^{\dagger} = p_{ji}^{*}\))

Consider a vector function \(\vec{u}(\vec{r})\) and two scalar functions \(\psi(\vec{r})\) and \(\phi(\vec{r})\). The unit vector \(\hat{n}\) is normal to the elementary surface \(dS\), \(dV\) is an infinitesimal volume, \(d\vec{l}\) is an infinitesimal line element, and \(\partial / \partial n\) denotes the partial derivative along \(\hat{n}\). Which of the following identities is/are correct?

A thin rod of uniform density and length \(2\sqrt{3}\, \text{m}\) is undergoing small oscillations about a pivot point. The time period of oscillation (\(T_m\)) is minimum when the distance of the pivot point from the center-of-mass of the rod is \(x_m\). Which of the following is/are correct? (Assume acceleration due to gravity \(g = 10\, \text{m/s}^2\))

An object executes simple harmonic motion along the x-direction with angular frequency \(\omega\) and amplitude \(a\). The speed of the object is 4 cm/s and 2 cm/s when it is at distances 2 cm and 6 cm, respectively, from the equilibrium position. Which of the following is/are correct?

For electric and magnetic fields, \(\vec{E}\) and \(\vec{B}\), due to a charge density \(\rho(\vec{r},t)\) and a current density \(\vec{J}(\vec{r},t)\), which of the following relations is/are always correct?

A linear operator \(\hat{O}\) acts on two orthonormal states of a system \(\psi_1\) and \(\psi_2\) as per the following: \[ \hat{O}\psi_1 = \psi_2, \hat{O}\psi_2 = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2) \] The system is in a superposed state defined by \[ \psi = \frac{1}{\sqrt{2}}\psi_1 + \frac{i}{\sqrt{2}}\psi_2 \] The expectation value of \(\hat{O}\) in the state \(\psi\) is:

Consider a one-dimensional infinite potential well of width \(a\). This system contains five non-interacting electrons, each of mass \(m\), at temperature \(T = 0 \, K\). The energy of the highest occupied state is:

A thin conducting square loop of side \(L\) is placed in the first quadrant of the \(xy\)-plane with one of the vertices at the origin. If a changing magnetic field \(\vec{B}(t) = B_0(52yt\,\hat{x} + xzt\,\hat{y} + 3y^2t^2\,\hat{z})\) is applied, where \(B_0\) is a constant, then the magnitude of the induced electromotive force in the loop is:

Consider \(N\) classical particles at temperature \(T\), each of which can have two possible energies 0 and \(\epsilon\). The number of particles in the lower energy level (\(N_0\)) and higher energy level (\(N_\epsilon\)) are related by (\(k_B\) is the Boltzmann constant):

The root mean square (rms) speeds of Hydrogen atoms at 500 K, \(V_{H}\), and Helium atoms at 2000 K, \(V_{He}\), are related as:

Two planets \(P_1\) and \(P_2\) having masses \(M_1\) and \(M_2\) revolve around the Sun in elliptical orbits, with time periods \(T_1\) and \(T_2\), respectively. The minimum and maximum distances of planet \(P_1\) from the Sun are \(R\) and \(3R\), respectively, whereas for planet \(P_2\), these are \(2R\) and \(4R\), respectively, where \(R\) is a constant. Assuming \(M_1\) and \(M_2\) are much smaller than the mass of the Sun, the magnitude of \(\dfrac{T_2}{T_1}\) is:

The intensity of the primary maximum in a two-slit interference pattern is given by \(I_2\), and the intensity of the primary maximum in a three-slit interference pattern is given by \(I_3\). Assuming the far-field approximation, same slit parameters, and same incident light intensity in both cases, \(I_2\) and \(I_3\) are related as:

The Boolean function \(\overline{P}(\overline{P} + Q)(Q + \overline{Q})\) is equivalent to: