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 $l$. [2]

$l$ 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 $\mathbf{b}$. [1]

\item Write down an equation of $l$ in vector form. [1]

$P$ is a point on $l$ such that $PA = 2OA$.

\item Find angle $POA$ giving your answer to 3 significant figures. [3]

$C$ 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 $p$. [3]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS ASFM 2020 Q2 [10]}}