7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre \(O\) and radius \(a\). The point \(A\) is the highest point on the hemisphere. A particle \(P\) is placed on the hemisphere at \(A\). It is then given an initial horizontal speed \(u\), where \(u ^ { 2 } = \frac { 1 } { 2 } ( a g )\). When \(O P\) makes an angle \(\theta\) with \(O A\), and while \(P\) remains on the hemisphere, the speed of \(P\) is \(v\).

1. Find an expression for \(v ^ { 2 }\).
2. Show that, when \(\theta = \arccos 0.9 , P\) is still on the hemisphere.
3. Find the value of \(\cos \theta\) when \(P\) leaves the hemisphere.
4. Find the value of \(v\) when \(P\) leaves the hemisphere. After leaving the hemisphere \(P\) strikes the table at \(B\).
5. Find the speed of \(P\) at \(B\).
6. Find the angle at which \(P\) strikes the table. \section*{Alternative Question 2:}
   1. Two light elastic strings \(A B\) and \(B C\) are joined at \(B\). The string \(A B\) has natural length 1 m and modulus of elasticity 15 N . The string \(B C\) has natural length 1.2 m and modulus of elasticity 30 N . The ends \(A\) and \(C\) are attached to fixed points 3 m apart and the strings rest in equilibrium with \(A B C\) in a straight line.
   Find the tension in the combined string \(A C\).