6. A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is hanging in equilibrium below \(O\) when it receives a horizontal impulse giving it a speed \(u\), where \(u ^ { 2 } = 3 g a\). The string becomes slack when \(P\) is at the point \(B\). The line \(O B\) makes an angle \(\theta\) with the upward vertical.

1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
   (9)
2. Show that the greatest height of \(P\) above \(B\) in the subsequent motion is \(\frac { 4 a } { 27 }\).
   (6)