2025 RI Promo Q10

With reference to the origin \(O\), the point \(A\) has position vector \(-\mathbf{i}-3\mathbf{j}+3\mathbf{k}\). The plane \(\pi\) and the line \(l\) have equations \[\mathbf{r}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\begin{pmatrix}-1\\3\\2\end{pmatrix}+\mu\begin{pmatrix}2\\3\\-4\end{pmatrix}\text{ and}\hspace{0.5em}\mathbf{r}=\begin{pmatrix}2\\1\\1\end{pmatrix}+\beta\begin{pmatrix}3\\t\\-2\end{pmatrix},\] respectively, where \(\lambda\), \(\mu\) and \(\beta\) are parameters and \(t\) is a real value.

1. It is given that \(l\) and \(\pi\) intersect at a point.
   1. Find the range of values of \(t\).[2]
   2. Find, in terms of \(t\), the coordinates of the point of intersection, \(B\), of \(l\) and \(\pi\).[2]
2. It is given that \(A\) lies on \(l\).
   1. Show that \(t=4\).[1]
   2. Find the position vector of the foot of perpendicular, \(F\), of \(A\) onto \(\pi\).[3]
   3. Find the cartesian equation of another line \(m\) through \(A\), that lies on the plane containing \(A\), \(B\) and \(F\), and \(m\) makes the same angle with \(\pi\) as \(l\) makes with \(\pi\).[4]