3 A particle \(P\) of mass \(m\) 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 \(O\). When \(P\) is hanging vertically below \(O\), it is given a horizontal speed \(u\). In the subsequent motion, \(P\) moves in a complete circle. When \(O P\) makes an angle \(\theta\) with the downward vertical, the tension in the string is \(T\). Show that $$T = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )$$ Given that the ratio of the maximum value of \(T\) to the minimum value of \(T\) is \(3 : 1\), find \(u\) in terms of \(a\) and \(g\). Assuming this value of \(u\), find the value of \(\cos \theta\) when the tension is half of its maximum value.