$(a)\;\large\frac{5}{3}\qquad(b)\;\large\frac{7}{5}\qquad(c)\;\large\frac{3}{2}\qquad(d)\;\large\frac{11}{9}$

$P \propto T^3$

$\implies P= KT^3$

For adiabatic process $\begin{align*}\frac{T^x}{P^{x-1}} = C \end{align*}$

where $C$ is a constant.

$\implies \begin{align*} \frac{T^x}{(KT^3)^{x-1}} = C \end{align*}$

$\implies \begin{align*} \frac{T^x}{(T^3)^{x-1}} = k^{x-1}. C\end{align*}$

$\implies \begin{align*} \frac{T^x}{T^{3x-3}} = k^{x-1}. C\end{align*}$

Let $K^{x-1}.C=A$

$\implies T^{3-2x } = A$

$\implies 3 - 2x = 0$

or $x = \frac{3}{2}$

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