A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.
(a) Use the equation of the trajectory given in the List of formulae (MF19), together with the condition \(y=0\), to establish an expression for the range \(R\) in terms of \(u, \theta\) and \(g\).

(b) Deduce an expression for the maximum height \(H\), in terms of \(u, \theta\) and \(g\).

It is given that \(R=\frac{4 H}{\sqrt{3}}\).
(c) Show that \(\theta=60^{\circ}\).

It is given also that \(u=\sqrt{40} \mathrm{~ms}^{-1}\).
(d) Find, by differentiating the equation of the trajectory or otherwise, the set of values of \(x\) for which the direction of motion makes an angle of less than \(45^{\circ}\) with the horizontal.