The temperature $(T)$ of one mole of an ideal gas varies with its volume $(V)$ as $T = -\alpha V^3 + \beta V^2$,where $\alpha$ and $\beta$ are positive constants. The maximum pressure of the gas during this process is ............

$A$ closed vessel contains a mixture of two diatomic gases $A$ and $B$. Molar mass of $A$ is $16$ times that of $B$ $(M_A = 16 M_B)$ and mass of gas $A$ contained in the vessel is $2$ times that of $B$ $(m_A = 2 m_B)$. Which of the following statements are true?
$(i)$ Average kinetic energy per molecule of $A$ is equal to that of $B$.
$(ii)$ Root mean square value of translational velocity of $B$ is four times that of $A$.
$(iii)$ Pressure exerted by $B$ is eight times that exerted by $A$.
$(iv)$ Number of molecules of $B$ in the cylinder is eight times that of $A$.

$A$ rocket is propelled by a gas which is initially at a temperature of $4000\, K$. The temperature of the gas falls to $1000\, K$ as it leaves the exhaust nozzle. The gas which will acquire the largest momentum while leaving the nozzle is:

An ideal gas expands adiabatically $(\gamma = 1.5)$. To reduce the root-mean-square (r.m.s.) velocity of the molecules $3$ times, the gas has to be expanded by a factor of: (in $\times$)

Which one of the following schematic graphs best represents the variation of $p V$ (in Joules) versus $T$ (in Kelvin) of one mole of an ideal gas? (The dotted line represents $p V=T$)

The figure shows a plot of $PV/T$ versus $P$ for $1.00 \times 10^{-3} \; kg$ of oxygen gas at two different temperatures.
$(a)$ What does the dotted plot signify?
$(b)$ Which is true: $T_{1} > T_{2}$ or $T_{1} < T_{2}$?
$(c)$ What is the value of $PV/T$ where the curves meet on the $y$-axis?
$(d)$ If we obtained similar plots for $1.00 \times 10^{-3} \; kg$ of hydrogen,would we get the same value of $PV/T$ at the point where the curves meet on the $y$-axis? If not,what mass of hydrogen yields the same value of $PV/T$ (for the low-pressure,high-temperature region of the plot)?
(Molecular mass of $H_{2} = 2.02 \; u$,of $O_{2} = 32.0 \; u$,$R = 8.31 \; J \; mol^{-1} K^{-1}$.)