2026 TJC P1 Q8

A plane \(\pi_1\) contains two vectors \(\begin{pmatrix}2\\1\\0\end{pmatrix}\) and \(\begin{pmatrix}0\\1\\2\end{pmatrix}\), and passes through the point \(A\) with coordinates \((1,0,1)\).

1. Show that the cartesian equation of \(\pi_1\) is \(x-2y+z=2\).[3]
2. The line \(l\) contains the point \(B\) with coordinates \((2,1,0)\) and is parallel to the vector \(\begin{pmatrix}1\\2\\k\end{pmatrix}\). Given that the acute angle between \(l\) and \(\pi_1\) is \(\theta\), where \(\sin\theta=\frac13\), find the possible values of \(k\).[4]
3. Given now that \(k=1\), find the coordinates of the point of intersection between \(l\) and \(\pi_1\).[3]
4. A plane \(\pi_2\) contains the point \(C\) with coordinates \((2,1,-3)\) and is parallel to \(\pi_1\). Find the exact distance between \(\pi_1\) and \(\pi_2\).[3]