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 \(O\). The particle \(P\) is moving in a complete vertical circle about \(O\). The points \(A\) and \(B\) are on the circle, at opposite ends of a diameter, and such that \(O A\) makes an acute angle \(\alpha\) with the upward vertical through \(O\). The speed of \(P\) as it passes through \(A\) is \(\frac{3}{2} \sqrt{ }(a g)\). The tension in the string when \(P\) is at \(B\) is four times the tension in the string when \(P\) is at \(A\).
(i) Show that \(\cos \alpha=\frac{3}{4}\).
(ii) Find the tension in the string when \(P\) is at \(B\).