The temperature of an ideal gas is increased from $$27^{\circ} \mathrm{C}$$ to $$927^{\circ} \mathrm{C}$$. The r.m.s. speed of its molecules becomes

A jar '$$\mathrm{P}$$' is filled with gas having pressure, volume and temperature $$\mathrm{P}, \mathrm{V}, \mathrm{T}$$ respectively. Another gas jar $$Q$$ filled with a gas having pressure $$2 \mathrm{P}$$, volume $$\frac{\mathrm{V}}{4}$$ and temperature $$2 \mathrm{~T}$$. The ratio of the number of molecules in jar $$\mathrm{P}$$ to those in jar $$Q$$ is

For a gas having '$$\mathrm{X}$$' degrees of freedom, '$$\gamma$$' is ($$\gamma=$$ ratio of specific heats $$=\mathrm{C_P / C_V}$$)

Two uniform brass rods $$A$$ and $$B$$ of length '$$l$$' and '$$2 l$$' and their radii '$$2 r$$' and '$$r$$' respectively are heated to same temperature. The ratio of the increase in the volume of $$\operatorname{rod} \mathrm{A}$$ to that of $$\operatorname{rod} \mathrm{B}$$ is