List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc- 2022

Consider a unit circle C in the xy plane, centered at the origin. The value of the integral \(\oint\)[(sin x - y)dx - (sin y - x)dy] over the circle C, traversed anticlockwise, is

A circular loop of wire with radius R is centered at the origin of the xy-plane. The magnetic field at a point within the loop is, \(\overrightarrow B(\rho,\phi,z,t)=k\rho^3t^3\hat{z}\), where k is a positive constant of appropriate dimensions. Neglecting the effects of any current induced in the loop, the magnitude of the induced emf in the loop at time t is

For a certain thermodynamic system, the internal energy U = PV and P is proportional to T². The entropy of the system is proportional to

Consider a particle trapped in a three-dimensional potential well such that U(x, y, z) = 0 for 0 ≤ x ≤ a, 0 ≤ y ≤ a, 0 ≤ z≤a and U(x, y, z) = ∞ everywhere else. The degeneracy of the 5^th excited state is

A hot body with constant heat capacity 800 //K at temperature 925 K is dropped gently into a vessel containing 1 kg of water at temperature 300 K and the combined system is allowed to reach equilibrium. The change in the total entropy $\bigtriangleup S$ is _____________ J/K (Round off to 1 decimal place).
[Take the specific heat capacity of water to be 4200 J/kg K. Neglect any loss of heat to the vessel and air and change in the volume of water.]

For an ideal gas, AB and CD are two isothermals at temperatures T₁ and T₂(\(T_1\gt T_2\)), respectively. AD and BC represent two adiabatic paths as shown in figure. Let V_A, V_B, V_C and V_D be the volumes of the gas at A, B, C and D respectively. If \(\frac{V_c}{V_B}=2\), then \(\frac{V_D}{V_A}=\)______.

A box contains a mixture of two different ideal monoatomic gases, 1 and 2, in equilibrium at temperature T. Both gases are present in equal proportions. The atomic mass for gas 1 is m, while the same for gas 2 is 2m. If the $rms$ speed of a gas molecule selected at random is $V_{rms} = x\sqrt{\frac{K_gT}{m}}$) then x is ________ (Round off to 2 decimal places).

For the given circuit, R = 125 Ω, R_L = 470 Ω, V_z = 9V, and \(I_z^{max}\)= 65 mA. The minimum and maximum values of the input voltage (\(V_i^{min}\ and\ V_i^{max}\)) for which the Zener diode will be in the 'ON' state are

A string of length L is stretched between two points x = 0 and x = L and the endpoints are rigidly clamped. Which of the following can represent the displacement of the string from the equilibrium position?

Consider the 2 X 2 matrix M= \(\begin{pmatrix} 0 & a \\ a & b \end{pmatrix}\), where a,b\(\gt\)0. Then,

A current density for a fluid flow is given by,
$ \overrightarrow{j} (x,y,z,t) = \frac{8e^t}{ (1+x^2+y^2+z^2)} \hat{x}$
At time t = 0, the mass density p(x, y, z, 0) = 1.
Using the equation of continuity, $p(1,1,1,1)$ is found to be ___________ (Round off to 2 decimal places).

A particle is executing simple harmonic motion with time period T. Let x, v an a denote the displacement, velocity and acceleration of the particle, respectively- at time t. Then,

The critical angle of a crystal is 30°. Its Brewster angle is _____________ degrees. (Round off to the nearest integer).

Consider an electron with mass m and energy E moving along the x-axis towards a finite step potential of height $U_0$, as shown in the figure. In region $1 (x < 0)$, the momentum of the electron is $p_1 = \sqrt{2mE}$. The reflection coefficient at the barrier is given by $R = (\frac{p_1-p_2}{p_1+p_2})^2$, where $p_2$ is the momentum in region 2. If, in the limit $E>> U_0$, R, then the integer n is ________.

The electric field associated with an electromagnetic radiation is given by E = a(1 + cosω₁t)cosω₂t. Which of the following frequencies are present in the field?

Let (r, \(\theta\)) denote the polar coordinates of a particle moving in a plane. If \(\hat{r}\) and \(\hat{\theta}\) represent the corresponding unit vectors, then

In the Compton scattering of electrons, by photons incident with wavelength λ,

The figure shows a section of the phase boundary separating the vapour (1) and liquid (2) states of water in the P-T plane. Here, C is the critical point. μ₁. ν₁ and s₁ are the chemical potential, specific volume and specific entropy of the vapour phase respectively, while μ₂. ν₂ and s₂ respectively denote the same for the liquid phase. Then

The Boolean expression Y = \(\overline{PQ}\)R+ Q\(\bar{R}\) + \(\bar{P}\)QR + PQR simplifies to

A charge $q$ is uniformly distributed over the volume of a dielectric sphere of radius $a$. If the dielectric constant $ \epsilon _r= 2$, then the ratio of the electrostatic energy stored inside the sphere to that stored outside is _________ . (Round off to I decimal place).

A linearly polarized light beam travels from origin to point A (1,0,0). At the point A, the light is reflected by a mirror towards point B (1, -1,0). A second mirror located at point B then reflects the light towards point C (1,-1,1). Let n(x, y, z) represent the direction of polarization of light at (x, y, z).

The work done in moving a $-5\muC$ charge in an electric field $ \overrightarrow{E} = (8r sin \theta \hat{r}+ + 4r cos \theta \hat{\theta} ) V/m$, from a point $A(r, \theta) = (10,\frac{\pi}{6})$ to a point $B(r,\theta) (10.\frac{\pi}{6})$ is _____ $mj$.

In an LCR series circuit, a non-inductive resistor of 150 Ω, a coil of 0.2 H inductance and negligible resistance, and a 30 $\mu F$ capacitor are connected across an ac power source of 220 V, 50 Hz. The power loss across the resistor is _________W (Round off to 2 decimal places).

For an n-type silicon, an extrinsic semiconductor, the natural logarithm of normalized conductivity (σ) is plotted as a function of inverse temperature. Temperature interval-I corresponds to the intrinsic regime, interval-II corresponds to saturation regime and interval-III corresponds to the freeze-out regime, respectively. Then

Consider the second order ordinary differential equation, y" +4y' +5y = 0. If y(0) = 0 and y'(0) = 1, then the value of у(π/2) is _________. (Round off to 3 decimal places).