If v be the linear velocity of rod after impact (upwards), $\omega$be the angular velocity of rod and J be the linear impulse at A during impact, then by

Impulse - Momentum Theorem, we have

$\begin{array}{l} \mathrm{J}=\mathrm{\Delta p}={\mathrm{p}}_{\mathrm{f}}-{\mathrm{p}}_{\mathrm{i}}\\ \Rightarrow \mathrm{J}=\mathrm{mv}-\left(-{\mathrm{mv}}_0\right)\\ \Rightarrow \mathrm{J}=\mathrm{m}\left(\mathrm{v}+{\mathrm{v}}_0\right) .........\left(1\right)\end{array}$

$\text{ Further by }\left(\begin{array}{l}\text{ Angular }\\ \text{ Impulse }\end{array}\right)-\left(\begin{array}{c}\text{ Angular }\\ \text{ Momentum }\\ \text{ Theorem }\end{array}\right),\text{ we have }$

$\mathrm{J}\left(\frac{\mathrm{\ell }}{2}\cos \mathrm{\theta }\right)=\mathrm{I\omega }=\frac{{\mathrm{m\ell }}^2}{12}\mathrm{\omega } ........\left(2\right)$

Since the collision is elastic, so at the point of impact e = 1.

$\Rightarrow \left(\begin{array}{c}\text{ Relative speed }\\ \text{ of approach }\end{array}\right)=\left(\begin{array}{c}\text{ Relative speed }\\ \text{ of separation }\end{array}\right)$

$\Rightarrow {\mathrm{v}}_0=\mathrm{v}+\frac{\mathrm{\ell }}{2}\mathrm{\omega cos}\mathrm{\theta } .......\left(3\right)$

Solving equations (1), (2) and (3), we get

$\mathrm{\omega }=\frac{6{\mathrm{v}}_0\cos \mathrm{\theta }}{\mathrm{\ell }\left(1+3{\cos }^2\mathrm{\theta }\right)}$

For $\omega$ to be maximum, we have

$\begin{array}{l} \frac{\mathrm{d\omega }}{\mathrm{d\theta }}=0\\ \Rightarrow \frac{6{\mathrm{v}}_0}{\mathrm{\ell }}\frac{\mathrm{d}}{\mathrm{d\theta }}\left(\frac{\cos \mathrm{\theta }}{1+3{\cos }^2\mathrm{\theta }}\right)=0\\ \Rightarrow \frac{\left(1+3{\cos }^2\mathrm{\theta }\right)(-\sin \mathrm{\theta })-\cos \mathrm{\theta }(-6\sin \mathrm{\theta cos}\mathrm{\theta })}{{\left(1+3{\cos }^2\mathrm{\theta }\right)}^2}=0\\ \Rightarrow 1+3{\cos }^2\mathrm{\theta }=6{\cos }^2\mathrm{\theta }\\ \Rightarrow 3{\cos }^2\mathrm{\theta }=1\\ \Rightarrow \cos \mathrm{\theta }=\frac{1}{\sqrt{3}}\\ \Rightarrow \ast =3\end{array}$