b
Moment of Inertia of complete disc about

\('O'\) point \({I_{total}} = \frac{{M{R^2}}}{2}\)

Radius of removed disc \(= R/4\)

Mass of removed disc \(= M/16\) [As \(M\) \( \propto \) \({{R^2}}\)]

\(M.I.\) of removed disc about its own axis \((O')\)

\( = \frac{1}{2}\frac{M}{{16}}{\left( {\frac{R}{4}} \right)^2} = \frac{{M{R^2}}}{{512}}\)

\(M.I\) of removed disc about \(O\)

\(\begin{array}{l}
{I_{remo\,ved\,disc\,}} = {I_{cm}} + m{x^2}\\
 = \frac{{M{R^2}}}{{512}} + \frac{M}{{16}}{\left( {\frac{{3R}}{4}} \right)^2} = \frac{{19\,M{R^2}}}{{512}}
\end{array}\)

\(M.I.\) of remaining disc 

\({I_{remaining}} = \frac{{M{R^2}}}{2} - \frac{{19}}{{512}}M{R^2} = \frac{{237}}{{512}}M{R^2}\)