5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
2. the value of \(\theta\) when \(P\) loses contact with the sphere,
3. the speed of \(P\) as it hits the table.
   (Total 13 marks)