Mechanics "proof" sum

Let the VELOCITY of the particle at the height h be u making angle θ with horizontal.clearly u=√(2gh).See that the maximum height of the projectile is 2h.so the particle will travel h distance upward and the corresponding range is 2h(given).treat this motion as a projectile. So (ucosθ)*T=2h; And T=2(usinθ)/g.put this value of T in the previous equation to get θ=45°.now calculate T.