2022 TJC Promo Q12

As part of a structure for a marble race, a plane \(p\) is placed at an angle \(\theta \) to the horizontal ground. The ground is taken to be the \(x-y\) plane and the line of intersection between \(p\) and the ground is the \(y\)-axis (see diagram).

1. Given that a point \(A\) with coordinates \(\left( 4,1,2 \right)\) lies on \(p\), find a vector equation of \(p\) in the scalar product form. Hence find the value of \(\theta \), leaving your answer in degrees correct to one decimal place.[4]
2. A straight rod is attached to \(p\) from \(A\) to a point \(B\) on the \(y\)-axis, as shown in the diagram. Given that \(AB=6\) units, find the position vector of \(B\).[4]

A marble is released from rest at \(A\) and it moves along the rod \(AB\) towards \(B\). It is observed that at time \(t\) seconds after the marble is released, the height of the marble above the ground is \(2-k{{t}^{2}}\) units where \(k\) is a positive constant less than \(1\).

3. Find in terms of \(k\),
   1. the time taken for the marble to reach \(B\).[1]
   2. the coordinates of marble when \(t=0.5\).[4]