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 $\mathbf { a }$.
\begin{enumerate}[label=(\alph*)]
\item Determine the shortest distance between the origin, $O$, and $l$.\\
$l$ is also perpendicular to the vector $\mathbf { b }$ where $\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }$.
\item Find a vector which is perpendicular to both $\mathbf { a }$ and $\mathbf { b }$.
\item Write down an equation of $l$ in vector form.\\
$P$ is a point on $l$ such that $P A = 2 O A$.
\item 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 $m$.
\item 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 $5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0$.\\
(a) Find the value of $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$.\\
(b) 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.
\end{enumerate}

\hfill \mbox{\textit{OCR FP1 AS 2021 Q2 [12]}}