\begin{enumerate}
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  \item With respect to a fixed origin $O$ the point $A$ has coordinates $( 3,6,5 )$ and the line $l$ has equation
\end{enumerate}

$$( \mathbf { r } - ( 12 \mathbf { i } + 30 \mathbf { j } + 39 \mathbf { k } ) ) \times ( 7 \mathbf { i } + 13 \mathbf { j } + 24 \mathbf { k } ) = \mathbf { 0 }$$

The points $B$ and $C$ lie on $l$ such that $A B = A C = 15$\\
Given that $A$ does not lie on $l$ and that the $x$ coordinate of $B$ is negative,\\
(a) determine the coordinates of $B$ and the coordinates of $C$\\
(b) Hence determine a Cartesian equation of the plane containing the points $A , B$ and $C$

The point $D$ has coordinates $( - 2,1 , \alpha )$, where $\alpha$ is a constant.\\
Given that the volume of the tetrahedron $A B C D$ is 147\\
(c) determine the possible values of $\alpha$

Given that $\alpha > 0$\\
(d) determine the shortest distance between the line $l$ and the line passing through the points $A$ and $D$, giving your answer to 2 significant figures.\\
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\hfill \mbox{\textit{Edexcel FP1 2023 Q7}}