A body of mass M moving at speed V _ 0 collides elastically with a...

Linear momentum conservation
(i) $y$-direction
$m{v}_{2}sin\theta = m{v}_{1}sin\theta$
$m{v}_{2} = m{v}_{1}$ ...(1)
(ii) $x$-direction
$m{v}_{0} = m{v}_{2}cos\theta + m{v}_{1}cos\theta$ ...(2)
from (1) and (2)
$m{v}_{0} = m{v}_{2}cos\theta + m{v}_{2}cos\theta$
$m{v}_{0} = 2m{v}_{2}cos\theta$ ...(3)
(iii) $e = 1$given
$\dfrac{{v}_{2} - {v}_{1}\cos 2\theta}{{v}_{0}cos\theta} = 1$
${v}_{2} - {v}_{1}\cos 2\theta = {v}_{0}cos\theta$ ...(4)
${v}_{2} - \dfrac{{m}_{2}2}{M}\cos 2\theta = {v}_{0}cos\theta$
${v}_{2}\left[1 - \dfrac{m}{M}\cos 2\theta\right] = {v}_{0}cos\theta$
$\dfrac{m{v}_{0}}{2mcos\theta}\left[1 - \dfrac{m}{M}\cos 2\theta\right] = {v}_{0}cos\theta$
$M - mcos2\theta = 2mcos2\theta$
$M = 2m\cos^{2}\theta + m\left(2\cos^{2}\theta - 1\right)$
$\dfrac{M}{m} = 4\cos^{2}\theta - 1$
{\left(\dfrac{M}{m}\right)}_\max = 4 - 1 = 3