A fixed quantity of an ideal gas can be expanded from an initial state to a certain volume through two different processes $\left(i\right) {\mathrm{PV}}^2=$constant and $\left(ii\right) \mathrm{P}={\mathrm{KV}}^2$ where $\mathrm{K}$ is a constant.

For the process $\left(i\right) {\mathrm{PV}}^2=$ constant

$\left(\frac{\mathrm{nRT}}{\mathrm{V}}\right){\mathrm{V}}^2=$ constant

$\therefore \mathrm{T}\alpha \frac{1}{ \mathrm{V}}$

for the process $\left(ii\right) \mathrm{P}={\mathrm{KV}}^2$

Therefore $\frac{nRT}{V}=k{V}^2 \Rightarrow T\infty V^3$

$\therefore$ On increasing volume, final temperature in process $\left(ii\right)$ is greater than that in process $\left(i\right)$ . So option $\left(2\right) is right .$

Now, for process ${\mathrm{PV}}^x=$constant, molar heat capacity, $\mathrm{C}=\frac{\mathrm{R}}{\gamma -1}-\frac{R}{1-x}$

$\therefore$ for process $\left(i\right) {\mathrm{C}}_1={\mathrm{C}}_{\mathrm{v}}-\mathrm{R}$

for process 

$$\begin{gathered} \left(ii\right) {\mathrm{C}}_2={\mathrm{C}}_{\mathrm{v}}-\frac{\mathrm{R}}{(-2-1)}={\mathrm{C}}_{\mathrm{v}}+\frac{\mathrm{R}}{3} \\ {\mathrm{C}}_2>{\mathrm{C}}_1 \mathrm{and} (\Delta \mathrm{T}{)}_{ii} \mathrm{is} \mathrm{positive} \mathrm{and} {\left(△\mathrm{T}\right)}_{\mathrm{i}} \mathrm{is} \mathrm{negative} . \\ So option \left(4\right) is right . \end{gathered}$$