7 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.

1. 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 \(g\).
2. Deduce an expression for the maximum height \(H\), in terms of \(u , \theta\) and \(g\).
   It is given that \(\mathrm { R } = \frac { 4 \mathrm { H } } { \sqrt { 3 } }\).
3. Show that \(\theta = 60 ^ { \circ }\).
   It is given also that \(u = \sqrt { 40 } \mathrm {~ms} ^ { - 1 }\).
4. 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.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.