If $C_1, C_2, C_3, ......$ represent the speeds of $n_1, n_2, n_3, ......$ molecules,then the root mean square speed is

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

At temperature $T \ K$,the $rms$ speed of $CO_2$ is $x \ cm/s$. At what temperature (in $K$) will the $rms$ speed of nitrous oxide $(N_2O)$ be $4x \ cm/s$ (in $T$)?

The $rms$ speed of hydrogen is $\sqrt{7}$ times the $rms$ speed of nitrogen. If the temperature of the gases is $T$,then:

At $T$ $(K)$,the $P$,$V$ and $u_{rms}$ of $1$ mole of an ideal gas were measured. The following graph is obtained. What is its slope $(m)$? ($x$-axis $= PV$; $y$-axis $= u_{rms}^2$,$M =$ Molar mass)

The most probable speed $(u_{mp})$ of $8 \ g$ of $H_2$ is $2 \times 10^2 \ ms^{-1}$. The kinetic energy (in $J$) of the same amount of $H_2$ gas is: