Two ideal gasses A and B have same pressure and volumes. A is expanded isothermally and B is expanded adiabatically to same final pressure, then select the correct alternative

$$\begin{gathered} \mathrm{Here} \mathrm{both} \mathrm{the} \mathrm{gases} \mathrm{has} \mathrm{same} \mathrm{initial} \mathrm{pressure} \mathrm{and} \mathrm{volume} \\ \mathrm{and} \mathrm{there} \mathrm{final} \mathrm{pressures} \mathrm{are} \mathrm{same}. \\ \mathrm{Assuming} \mathrm{initial} \mathrm{pressure} \mathrm{as} \mathrm{P} \mathrm{and} \mathrm{initial} \mathrm{volumes} \mathrm{as} \mathrm{V} \mathrm{where} \mathrm{as} \\ \mathrm{final} \mathrm{pressure} \mathrm{is} {\mathrm{P}}_2 \mathrm{and} \mathrm{final} \mathrm{volume} \mathrm{for} \mathrm{isothermal} \mathrm{process} \mathrm{is} {\mathrm{V}}_{\mathrm{A}} \mathrm{and} \\ \mathrm{for} \mathrm{adiabatic} \mathrm{process} \mathrm{it} \mathrm{is} {\mathrm{V}}_{\mathrm{B}} \end{gathered}$$

$$\begin{gathered} \mathrm{Then} \mathrm{PV}={\mathrm{P}}_2{\mathrm{V}}_{\mathrm{A}} \mathrm{for} \mathrm{isothermal} \mathrm{process} \\ {\mathrm{V}}_{\mathrm{A}}=\frac{\mathrm{PV}}{{\mathrm{P}}_2}\dots \dots \dots \left(a\right) \end{gathered}$$

$$\begin{gathered} {\mathrm{PV}}^{\mathrm{\gamma }}={\mathrm{P}}_2{\mathrm{V}}_{\mathrm{B}}^{\mathrm{\gamma }} \mathrm{for} \mathrm{adiabatic} \mathrm{process} \\ {\mathrm{V}}_{\mathrm{B}}^{\mathrm{\gamma }}=\frac{\mathrm{P}}{{\mathrm{P}}_2}{\mathrm{V}}^{\mathrm{\gamma }} \\ {\mathrm{V}}_{\mathrm{B}}={\left(\frac{\mathrm{P}}{{\mathrm{P}}_2}\right)}^{\frac{1}{\mathrm{\gamma }}}\mathrm{V}\dots \dots \left(\mathrm{b}\right) \end{gathered}$$

$\mathrm{dividing} \mathrm{equation} \left(\mathrm{a}\right) \mathrm{by} \left(\mathrm{b}\right) \mathrm{we} \mathrm{get}$

${\mathrm{V}}_{\mathrm{A}}={\left(\frac{\mathrm{P}}{{\mathrm{P}}_2}\right)}^{1-\frac{1}{\mathrm{\gamma }}}{\mathrm{V}}_{\mathrm{B}}$

$As\frac{ P}{P_2}>1$

$V_A>V_B$