The dimensional formula of angular impulse is _______.

The equations of two waves are given by \(y_{1}=5 \sin 2 \pi(x-v t) \,c m\,\) \(y_{2}=3 \sin 2 \pi(x-v t+1.5) \,c m\) These waves are simultaneously passing through a string. The amplitude of the resulting wave is.........\(cm\)

Match List-I with List-II.

List-I	List-II
A. \(\sin^2 \omega t\)	I. Periodic with time period \(T = \dfrac{\pi}{\omega}\) but not simple harmonic motion (SHM)
B. \(\sin^3(2\omega t)\)	II. Periodic with time period \(T = \dfrac{2\pi}{\omega}\) but Not SHM
C. \(\sin(\omega t) + \cos(\pi \omega t)\)	III. Periodic with time period \(T = \dfrac{\pi}{\omega}\) and SHM
D. \(\cos\omega t + \cos 2\omega t\)	IV. Non-periodic

Choose the correct answer from the options given below :

A gas mixture consists of \(8\) moles of argon and \(6\) moles of oxygen at temperature \(T\). Neglecting all vibrational modes, the total internal energy of the system is _______.

Two electrons each are fixed at a distance \('2d'\). A third charge proton placed at the midpoint is displaced slightly by a distance \(x ( x << d )\) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency : \(( m =\) mass of charged particle)

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in\(P-V\) diagram. The relation between the ratio \(\frac{V_a}{V_d}\) and the ratio \(\frac{V_b}{V_c}\) is _______.

Two ideal polyatomic gases at temperatures \(T _{1}\) and \(T _{2}\) are mixed so that there is no loss of energy. If \(F _{1}\) and \(F _{2}, m _{1}\) and \(m _{2}, n _{1}\) and \(n _{2}\) be the degrees of freedom, masses, number of molecules of the first and second gas respectively, the temperature of mixture of these two gases is

A sample of gas expands from \(V_1\) to \(V _2\). In which of the following, the work done will be greatest ?

Consider a tank made of glass (refractive index \(1.5\)) with a thick bottom. It is filled with a liquid of refractive index \(\mu \). A student finds that, irrespective of what the incident angle \(I\) (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of \(\mu \) is