The ratio of root mean square velocity to average velocity of gas molecules at a particular temperature is

The ratio of the most probable speed,average speed,and $RMS$ speed is given by:

Which of the following statement$(s)$ is (are) correct regarding the root mean square speed $(U_{rms})$ and average translational kinetic energy $(\varepsilon_{av})$ of a molecule in a gas at equilibrium?
$(1)$ $U_{rms}$ is doubled when its temperature is increased four times
$(2)$ $\varepsilon_{av}$ at a given temperature does not depend on its molecular mass
$(3)$ $U_{rms}$ is inversely proportional to the square root of its molecular mass
$(4)$ $\varepsilon_{av}$ is doubled when its temperature is increased four times

If the $RMS$ speed of nitrogen at a certain temperature is $3000 \ ms^{-1}$,the approximate kinetic energy of one mole of nitrogen at that temperature in $kJ$ is (assume nitrogen as ideal gas)

At the same temperature,which of the following gases will have the highest root mean square speed?

The rms velocity of oxygen molecules at $27^{\circ} C$ is around $800 \ m/s$. The rms velocity of methane molecules at $600 \ K$ temperature is around ...... (in $m/s$)