2023 RVHS P2 Q4

The line \({{l}_{1}}\) has equation \(\mathbf{r}=3\mathbf{k}+\lambda \left( \mathbf{i}+\mathbf{j} \right)\) where \(\lambda \) is a parameter. Referred to the origin \(O\), the points \(A\) and \(P\) have position vectors \(\mathbf{i}+4\mathbf{k}\). and \(-3\mathbf{i}+4\mathbf{j}+5\mathbf{k}\) respectively.

1. Find the cartesian equation of the plane \({{\pi }_{1}}\) containing the line \({{l}_{1}}\) and the point \(A\).[2]
2. The point \(Q\) is the foot of perpendicular of \(P\) on \({{l}_{1}}\). Find the position vector of \(Q\).[3]
3. The line \({{l}_{2}}\) passes through \(P\) and is parallel to the vector \(3\mathbf{i}+\mathbf{j}+m\mathbf{k}\) where \(m\) is a constant. Find the value of \(m\) given that \({{l}_{2}}\) is parallel to \({{\pi }_{1}}\).
   \(h\) denotes the distance between \({{l}_{2}}\) and \({{\pi }_{1}}\). Find the value of \(h\).[3]
4. The plane \({{\pi }_{2}}\) is parallel to \({{\pi }_{1}}\). The line \({{l}_{3}}\) passing through \(P\) and \(Q\) intersects \({{\pi }_{2}}\) at the point \(R\). The points \(P\), \(Q\) and \(R\) are such that \(PQ:PR=2:5\). Without finding the equation of \({{\pi }_{2}}\), find the possible value(s) of the distance between \({{\pi }_{1}}\) and \({{\pi }_{2}}\) in terms of \(h\).[2]