Which of the following statements about the kinetic theory of gases is wrong?

Read the given statements and decide which is/are correct on the basis of the kinetic theory of gases:
$(I)$ Energy of one molecule at absolute temperature $T = 0 \ K$ is zero.
$(II)$ $r.m.s.$ speeds of different gases are the same at the same temperature.
$(III)$ For one gram of all ideal gases,kinetic energy is the same at the same temperature.
$(IV)$ For one mole of all ideal gases,mean kinetic energy is the same at the same temperature.

$STATEMENT-1$: The total translational kinetic energy of all the molecules of a given mass of an ideal gas is $1.5$ times the product of its pressure and its volume. Because
$STATEMENT-2$: The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

Two gases of equal mass are in thermal equilibrium. If $P_a, P_b$ and $V_a, V_b$ are their respective pressures and volumes,then which relation is true?

For the equation $\beta = - \frac{1}{V} \left( \frac{dV}{dP} \right)$, identify the correct graph of $\beta$ versus $P$ for an ideal gas at constant temperature.

$2.8 \ g$ of nitrogen gas $(N_2)$ is in a vessel at a temperature of $127^{\circ} C$. The amount of heat energy required to increase the rms speed of the nitrogen molecules by $41.4 \%$ is $(R = 8.31 \ J \ mol^{-1} \ K^{-1})$. (in $J$)