2
The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).

1. Determine the shortest distance between the origin, \(O\), and \(l\).
   \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
3. Write down an equation of \(l\) in vector form.
   \(P\) is a point on \(l\) such that \(P A = 2 O A\).
4. Find angle \(P O A\) giving your answer to 3 significant figures.
   \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
5. Find the value of \(p\). \section*{In this question you must show detailed reasoning.} You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).
6. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
7. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.