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A nuclear reactor starts producing a radioactive nuclide \(X\) from \(t = 0\), at a constant rate of \(\alpha\) per second. Each decay of \(X\) produces energy \(E_{0}\), which is utilized to heat a liquid of mass \(m\) and specific heat \(s\). Assuming no heat loss from the liquid and taking \(\lambda\) as the decay constant of \(X\), the rate of increase in the temperature of the liquid is:

A beam of polychromatic light passes through a thin prism of prism angle \(6^\circ\). The refractive index of the material of the prism varies with wavelength \((\lambda)\) as \(n(\lambda) = \alpha \lambda + \dfrac{\beta}{\lambda^{2}}\) where \(\alpha = 3 \, \mu \text{m}^{-1}\) and \(\beta = 0.096 \, \mu \text{m}^{2}\). If \(\lambda_{\min}\) is the wavelength at which the angle of minimum deviation \(D_{m}\) is smallest, then the correct value of \(D_{m}\) at \(\lambda_{\min}\) is:

A particle of mass \(m\), and angular momentum \(l\) is moving in a circular orbit of radius \(r_{0}\) under the influence of an attractive force \( \vec{F}(r) = - \dfrac{k}{r^{2}} \, \hat{r} \). Keeping its angular momentum unchanged, the particle is displaced radially by small distance \(\delta r \ll r_{0}\), due to which its radial distance varies periodically.
The corresponding time period is:

Consider two isosceles prisms \( 1 \) and \( 2 \) with prism angles \( A_1 \) and \( A_2 \) and refractive indices \( n_1 \) and \( n_2 \), respectively, as shown in the figure. The faces \( a_1 b_1 \) and \( a_2 b_2 \) are parallel to each other and perpendicular to the mirror \( M \). If a ray of light is incident on the face \( a_1 c_1 \) and emerges from the face \( a_2 c_2 \), then the correct statement(s) is/are:

In a vacuum chamber, a particle of charge \( 1 \,\mu\text{C} \) and mass \( 1 \,\text{mg} \) is projected with a velocity \( (\hat{i} + 2\hat{j}) \,\text{m}\,\text{s}^{-1} \) from the \( XZ \) plane at time \( t = 0 \) in an electric field of \( 1 \hat{i} \,\text{V}\,\text{m}^{-1} \). At \( t = 0.2 \,\text{s} \), the electric field is switched off and a magnetic field of \( 6 \hat{j} \,\text{T} \) is switched on. The acceleration due to gravity is \( -10 \hat{j} \,\text{m}\,\text{s}^{-2} \). Correct option(s) is are:

Two charges \( Q_1 = q \) and \( Q_2 = m q \) are placed at the time points \( P_1(a, b) \) and \( P_2(m a, m b) \), respectively, in the \( XY \) plane, where \( a, b \neq 0 \) and \( m \neq 0, 1 \). If \( V_1 \) is the potential at a point in the \( XY \) plane due to charge \( Q_1 \) and \( V_2 \) is the potential at that point due to charge \( Q_2 \). Correct statement(s) for the points at which \( |V_1| = |V_2| \) is/are:

Consider an electric dipole comprising two charges \( +q \) and \( -q \) each with mass \( m \), separated by a fixed distance \( d \) and initially at rest with its dipole moment pointing along \( \hat{i} \). A uniform electric field \( {E}\,\hat{j} \) is turned on at time \( t = 0 \) and it is turned off at \( t = t_f \), when the dipole moment makes an angle \( \theta_f \) with \( \hat{i} \). Neglecting any sources of energy loss, correct option(s) is/are:

Ten moles of an ideal monoatomic gas, initially in state \( a \) at atmospheric pressure and temperature \( T_a = 27^\circ C \), is enclosed in a metal cylinder of volume \( V_0 \) fitted with a frictionless piston. The gas is suddenly compressed to state \( b \) with volume \( \frac{V_0}{3} \). Now, keeping the piston stationary, the cylinder is submerged in a water bath of temperature \( 11^\circ C \) until the gas reaches the temperature of the water bath, which is denoted as state \( c \). Finally, while still in the water bath, the piston is brought slowly to its initial position, which is denoted as state \( f \). If \( R \) is universal gas constant, then the correct option(s) is/are: [Given: \( 9^{1/3} = 2.08 \)]

Two thin wires, Wire-1 of diameter \( 0.650 \, \text{mm} \) and Wire-2 of unknown diameter \( d \) are given. To obtain the value of \( d \), the diameters of the two wires are measured with a screw gauge. The screw gauge has a pitch of \( 0.5 \, \text{mm} \) and there are \( 100 \) divisions on the circular scale (CS). The smallest division on the linear scale (LS) is \( 0.5 \, \text{mm} \). The table shows the readings of LS and CS for the measurements. The value of \( d \) (in \( \mu \text{m} \)) is ______.

In a single slit diffraction experiment, a slit of width \( (0.016 \pm 0.002) \, \text{mm} \) is used to measure the wavelength of a monochromatic light source. In the diffraction pattern, the angular distance between the central maximum and first minimum is measured to be \( (2^\circ \pm 40') \). The value of the fractional error in the measurement of wavelength is _______. [Given: \( \sin(2^\circ) = 0.035 \)]

As shown in the figure, a ray \( AB \) of unpolarized light enters from water of refractive index \( n_w = \frac{4}{3} \) into a medium of refractive index \( n_p = \frac{4}{\sqrt{3}} \) after passing through a glass plate of refractive index \( n_g = 1.5 \) and a layer of water. At a particular incident angle \( i \) the reflected ray \( CD \) is polarized in the direction as shown in the figure. The value of \( i \) (in degrees) is ________.

As shown in the figure, the resistance of a galvanometer \( G \) can be found by the half-deflection method. Here the resistance \( R_2 \) is adjusted such that when the key \( K \) is closed the deflection in the galvanometer becomes half of the value as compared to when \( K \) is open. Half-deflection is obtained at \( R_2 = 4\,\Omega \) and thus the galvanometer resistance is found to be \( 6\,\Omega \). In this half-deflection condition the current (in \( \text{mA} \)) through the resistor \( R_1 \) is ________.

In a new system of units, the units of mass, length, time and current are \( 5\,\text{kg} \), \( 5\,\text{m} \), \( 5\,\text{s} \) and \( 5\,\text{A} \), respectively. If \( \mu_0 \) and \( \varepsilon_0 \) are the permeability and permittivity of free space, respectively, then in this new system of units, the magnitude of one SI unit of \( \sqrt{\mu_0 / \varepsilon_0} \), is ________.

A container of height \( 2\,\text{m} \), length \( 2\,\text{m} \) and breadth \( 1\,\text{m} \) is made of insulating vertical walls and two large area horizontal metal plates (\( M_1 \) and \( M_2 \)) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area \( \sqrt{10}\,\text{cm}^2 \) near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant \( \varepsilon_r = 15 \) and the right chamber is empty (\( \varepsilon_r = 1 \)). At time \( t = 0 \), the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has \( \varepsilon_r = 1 \) and is maintained at atmospheric pressure. The schematic of the container at a time \( t > 0 \) is shown in the figure. [Given: acceleration due to gravity is \( 10\,\text{m s}^{-2} \).]

The height (in \( \text{m} \)) of the liquid in left chamber at \( t = 500\,\text{s} \) is ________.

A container of height \( 2\,\text{m} \), length \( 2\,\text{m} \) and breadth \( 1\,\text{m} \) is made of insulating vertical walls and two large area horizontal metal plates (\( M_1 \) and \( M_2 \)) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area \( \sqrt{10}\,\text{cm}^2 \) near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant \( \varepsilon_r = 15 \) and the right chamber is empty (\( \varepsilon_r = 1 \)). At time \( t = 0 \), the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has \( \varepsilon_r = 1 \) and is maintained at atmospheric pressure. The schematic of the container at a time \( t > 0 \) is shown in the figure. [Given: acceleration due to gravity is \( 10\,\text{m s}^{-2} \).]

The difference in the capacitance (in \( \text{F} \)) between the metal plates at \( t = 0 \) and that at \( t = 500\,\text{s} \) is \( (8 - n)\,\varepsilon_0 \), where \( \varepsilon_0 \) is the permittivity of free space. The value of \( n \) is ________.

A uniform circular disk of radius \( 0.2 \, \text{m} \) and mass \( 1 \, \text{kg} \) is pivoted at its top point C such that it can rotate freely around C in the XY plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass \( 20 \, \text{g} \), travelling along negative \( x \) direction in the XY plane with speed \( 100 \, \text{ms}^{-1} \), hits the circumference of the disk at a point P. After collision the particle moves along negative \( y \) direction at a speed of \( 90 \, \text{ms}^{-1} \).
[Given: the acceleration due to gravity \( (g) = -10 \, \hat{\jmath} \, \text{ms}^{-2} \)]

After the collision the disk starts to rotate around point C in the XY plane. The maximum change in the height (in \( \text{m} \)) of its centre O is _______.

A uniform circular disk of radius \( 0.2 \, \text{m} \) and mass \( 1 \, \text{kg} \) is pivoted at its top point C such that it can rotate freely around C in the XY plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass \( 20 \, \text{g} \), travelling along negative \( x \) direction in the XY plane with speed \( 100 \, \text{ms}^{-1} \), hits the circumference of the disk at a point P. After collision the particle moves along negative \( y \) direction at a speed of \( 90 \, \text{ms}^{-1} \).
[Given: the acceleration due to gravity \( (g) = -10 \, \hat{\jmath} \, \text{ms}^{-2} \)]

Amount of energy loss (in \( \text{J} \)) in the collision is _______.