The molar heat capacity in a process of a monoatomic gas, if it does a work of ‘Q/4’ when a heat of Q is supplied to it is

$\begin{array}{l}\mathrm{dU}={\mathrm{C}}_{\mathrm{V}}\mathrm{dT}=\left(\frac{3}{2}\mathrm{R}\right)\mathrm{dT}\\ \text{ or }\mathrm{dT}=\frac{2\left(\mathrm{dU}\right)}{3\mathrm{R}} ....\left(\mathrm{i}\right)\end{array}$
From first law of thermodynamics
$\mathrm{dU}=\mathrm{dQ}-\mathrm{dW}=\mathrm{Q}-\frac{\mathrm{Q}}{4}=\frac{3\mathrm{Q}}{4}.$
Now molar heat capacity $\mathrm{C}=\frac{\mathrm{dQ}}{\mathrm{dT}}=\frac{\mathrm{Q}}{\frac{2\left(\mathrm{dU}\right)}{3\mathrm{R}}}=\frac{\mathrm{Q}\left(3\mathrm{R}\right)}{2\left(\frac{3\mathrm{Q}}{4}\right)}=2\mathrm{R}$