Under isothermal conditions, two soap bubbles of radii $r_1$ and $r_2$ coalesce to form a big drop. The radius of the big drop is

The frequencies for series limit of Balmer and Paschen series are ' $\mathrm{V}_1$ ' and ' $\mathrm{V}_3$ ' respectively. If frequency of first line of Balmer series ' $\mathrm{V}_2$ ' then the relation between $V_1, V_2$ and $V_3$ is

Two cylinders A and B fitted with piston contain equal amount of an ideal diatomic gas at temperature T K. The piston of cylinder A is free to move while that of cylinder $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise of temperature of the gas in A is $\mathrm{dT}_{\mathrm{A}}$, then the rise in temperature of the gas in B is $\left(\gamma=\frac{C_p}{C_v}\right)$

In a certain 2 -inputs logic gate, when inputs $\mathrm{A}=0$ and $\mathrm{B}=0$, then output $\mathrm{C}=1$. And also when inputs $\mathrm{A}=0, \mathrm{~B}=1$, then again output $\mathrm{C}=1$. The gate must be