If the temperature of a gas is increased from $37\,^oC$ to $927\,^oC$,then its original root mean square speed will become ...........

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

Calculate the root mean square $(rms)$ speed of ozone gas in a closed vessel at a temperature of $20 \, ^\circ C$ and a pressure of $82 \, cm \, Hg$.

For a gaseous state,if the most probable speed is denoted by $C^*$,average speed by $\overline{C}$,and root mean square speed by $C$,then for a large number of molecules,the ratio of these speeds is:

Case $-1:$ An ideal gas of molecular weight $M$ at temperature $T$.
Case $-2:$ Another ideal gas of molecular weight $2M$ at temperature $T/2$.
Identify the correct statement in the context of the above two cases.

The $RMS$ velocity $(u_{rms})$ of one mole of an ideal gas was measured at different temperatures and the following graph is obtained. What is the slope $(m)$ of the straight line?
($x$-axis $= T$ $(K)$; $y$-axis $= (u_{rms})^2$; $M =$ molar mass; $R =$ gas constant)