2020 YIJC P2 Q4

The plane \({{\pi }_{1}}\) has equation \(\mathbf{r}\centerdot \left( \begin{matrix}2 \\5 \\1 \end{matrix} \right)=11\). The plane \({{\pi }_{2}}\) passes through the points \(A\left( 1,-1,2 \right)\), \(B\left( 2,3,4 \right)\) and \(C\left( 3,4,3 \right)\).

1. Find a cartesian equation of \({{\pi }_{2}}\).[3]
2. Find a vector equation of the line \(l\) where \({{\pi }_{1}}\) and \({{\pi }_{2}}\) meet.[2]
3. \(D\) is a general point on \(l\). Find an expression for the square of the distance \(AD\). Hence, or otherwise, find the coordinates of the point on \(l\) which is nearest to \(A\).[5]
4. The plane \({{\pi }_{3}}\) is parallel to \({{\pi }_{1}}\) and contains the point \(E\left( 1,k,3 \right)\). Given that the distance between \({{\pi }_{1}}\) and \({{\pi }_{3}}\) is \(\frac{16}{\sqrt{30}}\), find the possible values of \(k\).[3]