2024 IGCSE Additional Mathematics May/June 0606/12 Q9

In this question, all distances are in metres and time, \(t\), is in seconds.

A particle \(P\) moves with a speed of \(14.5\) parallel to the vector \(\begin{pmatrix}-20\\21\end{pmatrix}\).

1. Find the velocity vector of \(P\).[2]

Initially, \(P\) has position vector \(\begin{pmatrix}3\\5\end{pmatrix}\).

2. Write down the position vector of \(P\) at time \(t\).[2]

A second particle \(Q\) has position vector \(\begin{pmatrix}-1\\3\end{pmatrix}+\begin{pmatrix}-5\\7.5\end{pmatrix}t\) at time \(t\).

3. Find, in terms of \(t\), the distance between \(P\) and \(Q\) at time \(t\). Simplify your answer.[4]
4. Hence show that \(P\) and \(Q\) never collide.[2]