8. With respect to a fixed origin $O$, the lines $l _ { 1 }$ and $l _ { 2 }$ are given by the equations

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 
2 \\
- 3 \\
4
\end{array} \right) + \lambda \left( \begin{array} { r } 
- 1 \\
2 \\
1
\end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 
2 \\
- 3 \\
4
\end{array} \right) + \mu \left( \begin{array} { r } 
5 \\
- 2 \\
5
\end{array} \right)$$

where $\lambda$ and $\mu$ are scalar parameters.
\begin{enumerate}[label=(\alph*)]
\item Find, to the nearest $0.1 ^ { \circ }$, the acute angle between $l _ { 1 }$ and $l _ { 2 }$

The point $A$ has position vector $\left( \begin{array} { l } 0 \\ 1 \\ 6 \end{array} \right)$.
\item Show that $A$ lies on $l _ { 1 }$

The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $X$.
\item Write down the coordinates of $X$.
\item Find the exact value of the distance $A X$.

The distinct points $B _ { 1 }$ and $B _ { 2 }$ both lie on the line $l _ { 2 }$\\
Given that $A X = X B _ { 1 } = X B _ { 2 }$
\item find the area of the triangle $A B _ { 1 } B _ { 2 }$ giving your answer to 3 significant figures.

Given that the $x$ coordinate of $B _ { 1 }$ is positive,
\item find the exact coordinates of $B _ { 1 }$ and the exact coordinates of $B _ { 2 }$\\

\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}
\end{enumerate}

\hfill \mbox{\textit{Edexcel C4 2014 Q8 [15]}}