We know that $PV = n RT$ and are given $P = kV^k$.

$\Rightarrow PV = nRT = kV^{k+1}$

Differentiating, we get $nR dT = (k+1)kV^n dV \rightarrow (k+1) PdV = nRdT$

Now, $dQ = dU + pDV$

$\Rightarrow nCdT = nC_vdT + \large\frac{nRdT}{k+1}$

$\Rightarrow C = C_v + R/(k+1)$

$\Rightarrow C = 3R/2 + R/(k+1)$