$A$ sample of gas is at $0^{\circ}C$. To what temperature must it be raised in order to double the $r.m.s.$ speed of the molecules? (in $^{\circ}C$)

At which temperature is the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $47^{\circ} C$ (in $\,K$)?

The velocity of the molecules of a gas at temperature $120\,K$ is $v$. At what temperature in $K$ will the velocity be $2v$?

At what temperature does an oxygen molecule have the same r.m.s. velocity as a hydrogen molecule at $20 \ K$ (in $K$)?

Let $\bar{v}$,${v_{rms}}$ and ${v_p}$ respectively denote the mean speed,root mean square speed,and most probable speed of the molecules in an ideal monoatomic gas at absolute temperature $T$. The mass of a molecule is $m$. Then

Column-$I$ represents the formula for ${v_{rms}}$ and Column-$II$ represents the corresponding condition (phenomena). Match them correctly:

Column-$I$	Column-$II$
$(a)$ ${v_{rms}} = \sqrt {\frac{3P}{\rho}}$	$(i)$ For $1 \text{ mole ideal gas}$
$(b)$ ${v_{rms}} = \sqrt {\frac{3RT}{M_0}}$	$(ii)$ For one molecule of gas
$(c)$ ${v_{rms}} = \sqrt {\frac{3{k_B}T}{m}}$	$(iii)$ On the basis of kinetic theory