$\mathbf { 7 } \quad$ The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]$, where $q$ is an integer.

The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]$.

The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.
\begin{enumerate}[label=(\alph*)]
\item Show that $q = 4$ and find the coordinates of $P$.
\item Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular.
\item The point $A$ lies on the line $l _ { 1 }$ where $\lambda = 1$.
\begin{enumerate}[label=(\roman*)]
\item Find $A P ^ { 2 }$.
\item The point $B$ lies on the line $l _ { 2 }$ so that the right-angled triangle $A P B$ is isosceles.

Find the coordinates of the two possible positions of $B$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA C4 2012 Q7 [12]}}