One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and is held with the string taut at the point \(A\). At \(A\) the string makes an angle \(\theta\) with the upward vertical through \(O\). The particle is projected perpendicular to the string in a downward direction from \(A\) with a speed \(u\). It moves along a circular path in the vertical plane.

When the string makes an angle \(\alpha\) with the downward vertical through \(O\), the speed of the particle is \(2 u\) and the magnitude of the tension in the string is 10 times its magnitude at \(A\).

It is given that \(u=\sqrt{\frac{2}{3} g a}\).
(a) Find, in terms of \(m\) and \(g\), the magnitude of the tension in the string at \(A\).

(b) Find the value of \(\cos \alpha\).