N2023 Specimen Paper P1 Q10

The point \(A\) has coordinates \(\left( 0,\ 2,\ -3 \right)\). The line \(l\) has equation \(\frac{x}{2}=\frac{y+1}{-3}=\frac{z-2}{1}\).

1. Find the cartesian equation of the plane \(\pi \) which contains \(A\) and is perpendicular to \(l\).[3]
2. Hence, or otherwise, find the coordinates of the point \(P\) on \(l\) which is closest to \(A\).[3]
3. The line \(m\) passes through the point with coordinates \(\left( 4,\ -5,\ 10 \right)\) and \(P\). The line \(n\) lies in the same plane as \(l\) and \(m\). Find a cartesian equation for \(n\) if \(n\) is the reflection of the line \(m\) in the line \(l\).[6]