Two immiscible liquids of refractive indices \(\frac{8}{5}\) and \(\frac{3}{2}\) respectively are put in a beaker as shown in the figure. The height of each column is \(6 \mathrm{~cm}\). A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is \(\frac{\alpha}{4} \mathrm{~cm}\). The value of \(\alpha\) is__________.

For the circuit shown, the value of current at time \({t}=3.2\, {s}\) will be ...... \({A}\). [Voltage distribution \(V(t)\) is shown by Fig. \((1)\) and the circuit is shown in Fig. \((2)\)]

A light whose electric field vectors are completely removed by using a good Polaroid, allowed to incident on the surface of the prism at Brewster's angle. Choose the most suitable option for the phenomenon related to the prism.

A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same \(P, V, T\). Heating is started from left side until pressure changes to \(\dfrac{27P}{8}\). If initial volume of each compartment was \(9\) litres then the final volume in right-hand side compartment is __________ litres. (for this ideal gas \(C_P/C_V = 1.5\))

The sum of two forces \(\overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) is \(\overrightarrow{\mathrm{R}}\) such that \(|\overrightarrow{\mathrm{R}}|=|\overrightarrow{\mathrm{P}}| .\) The angle \(\theta\) (in degrees) that the resultant of \(2 \overrightarrow{\mathrm{P}}\) and \(\overrightarrow{\mathrm{Q}}\) will make with \(\overrightarrow{\mathrm{Q}}\) is

Match List\(-I\) with List\(-II\) 

List\(-I\)	List\(-II\)
\((A)\) \(AC\) generator	\((I)\) Detects the presence of current in the circuit
\((B)\) Galvanometer	\((II)\) Converts mechanical energy into electrical energy
\((C)\) Transformer	\((III)\) Works on the principle of resonance in \(AC\) circuit
\((D)\) Metal detector	\((IV)\) Changes an alternating voltage for smaller or greater value

Choose the correct answer from the options given below

A thermodynamic system is taken through the cyclic process ABC as shown in the figure (P-V graph : A(2, 100), B(5, 300), C(5, 100). The total work done by the system during the cycle ABC is ___________ J.

A black coloured solid sphere of radius \(R\) and mass \(M\) is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature \(T_0\). The initial temperature of the sphere is \(3T_0\). If the specific heat of the material of the sphere varies as \(\alpha T^3\) per unit mass with the temperature \(T\) of the sphere, where \(\alpha \) is a constant, then the time taken for the sphere to cool down to temperature \(2T_0\) will be ( \(\sigma \) is Stefan Boltzmann constant)

Two projectiles are fired from ground with same initial speeds from same point at angles \(\left(45^{\circ}+\alpha\right)\) and \(\left(45^{\circ}-\alpha\right)\) with horizontal direction. The ratio of their times of flights is

A sphere of relative density \(\sigma\) and diameter \(D\) has concentric cavity of diameter \(d\). The ratio of \(\frac{D}{d}\), if it just floats on water in a tank is _______.