Linear momentum conservation

            (i)         y-direction

                        mv₂sinθ = mv₁sinθ

                        mv₂ = mv₁                                             …….(1)

            (ii)         x-direction

                        mv₀ = mv₂cosθ + mv₁cosθ                     …….(2)

            from (1) and (2)

                        mv₀ = mv₂cosθ + mv₂cosθ

                        mv₀ = 2mv₂cosθ                                    …….(3)

            (iii)        e = 1 given

                        $\frac{{\mathrm{v}}_2-{\mathrm{v}}_1\cos 2\mathrm{\theta }}{{\mathrm{v}}_0\mathrm{cos\theta }}=1$

                        v₂ – v₁cos₂θ = v₀cosθ                            …….(4)

                        ${\mathrm{v}}_2-\frac{{\mathrm{mv}}_2}{\mathrm{M}}\cos 2\mathrm{\theta }={\mathrm{v}}_0\mathrm{cos\theta }$

                        ${\mathrm{v}}_2[1-\frac{\mathrm{m}}{\mathrm{M}}\cos 2\mathrm{\theta }]={\mathrm{v}}_0\mathrm{cos\theta }$

                        $\frac{{\mathrm{mv}}_0}{2\mathrm{mcos\theta }}[1-\frac{\mathrm{m}}{\mathrm{M}}\cos 2\mathrm{\theta }]={\mathrm{v}}_0\mathrm{cos\theta }$

                        M – mcos2θ = 2mcos2θ ;

                        M = 2mcos²θ = m(2cos²θ – 1)

                       $\frac{\mathrm{M}}{\mathrm{m}}=4{\cos }^2-1$

                     ${\left(\frac{\mathrm{M}}{\mathrm{m}}\right)}_{\max }=4-1=3$