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).
(i)
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]
(ii)
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 $1$.
(iii)
Find in terms of $k$,
(a) the time taken for the marble to reach $B$.
[1]
(b) the coordinates of marble when $t=0.5$.
[4]
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