2022 NJC Promo Q10

A player controls an avatar in an exploration game to find treasures. In one of the stages, the avatar scales a cliff. The sloping surfaces of the cliff can be modelled by plane \({{p}_{1}}\) with equation \(x-5z=-2\) and plane \({{p}_{2}}\) with equation \(4x+y-nz=30\) where \(n\) is a constant. The top surface of the cliff can be modelled by plane \({{p}_{3}}\) with equation \(x-12z=-450\) which contains the point \(T\left( 30,\,30,\,40 \right)\), where a treasure is located. The figure below shows a side view of the cliff.

Locations of points \(\left( x,\,y,\,z \right)\) are defined relative to a point \(\left( 0,\,0,\,0 \right)\) at the foot of the cliff and it is assumed that the avatar travels from a point to another using the shortest path. The avatar expends \(1\) unit of stamina for every \(2\) units travelled.

1. The ground in the game can be modelled by the plane with equation \(z=0\). Given that \({{p}_{2}}\) is steeper than \({{p}_{1}}\) relative to the ground, find the range of values of \({{n}^{2}}\).[4]

It is given further that \(T\left( 30,\,30,\,40 \right)\) also lies in \({{p}_{2}}\).

2. Show that \(n=3\).[1]
3. The boundary where \({{p}_{1}}\) and \({{p}_{2}}\) meet can be modelled by the line \(l\). Verify that a vector equation of boundary \(l\) is \(\mathbf{r}=\left( \begin{matrix} 8 \\ 4 \\ 2 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 5 \\ -17 \\ 1 \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).[2]
4. The player resumes the game with the avatar on \({{p}_{1}}\). The avatar moves towards \(B\), a point on boundary \(l\) such that \(BT\) is minimum. Find the exact position vector of \(B\). Hence, find the total stamina expended by the avatar to travel from \(B\) to \(T\).[5]
5. From the point \(T\), the avatar travels along a path parallel to \(\left( \begin{matrix} 10 \\ 3a \\ 9a \\ \end{matrix} \right)\), where \(a\) is a constant. Another treasure is located at \(R\left( 90,\,0,\,45 \right)\) on \({{p}_{3}}\). Determine whether the avatar will eventually reach this treasure.[5]