b
Angular momentum, \(\overrightarrow{ L }=\overrightarrow{ r } \times m \overrightarrow{ v }\)

\(\Rightarrow|\overrightarrow{ L }|= rmv \sin \theta\)

At maximum point, velocity is \(v = v \cos \theta= v \cos (30)=\frac{\sqrt{3} v }{2}\) only (direction: towards horizontal) and no vertical velocity is present.

The maximum height reached will be

\(h=\frac{v^{2} \sin ^{2} \theta}{2 g}\)

\(\Rightarrow h =\frac{ v ^{2} \sin ^{2}\left(30^{\circ}\right)}{2 g }\)

\(\Rightarrow h =\frac{ v ^{2}}{8 g }\)

\(L = rmv \sin \theta\)

\(\Rightarrow L = mvh\)

\(\Rightarrow L = m \times \frac{\sqrt{3} v }{2} \times \frac{ v ^{2}}{8 g }\)

\(\Rightarrow L =\frac{\sqrt{3} mv ^{3}}{16 g }\)