2025 IGCSE Additional Mathematics October/November 0606/21 Q12

In this question, the \(x\)- and \(y\)-directions are east and north respectively. The units are metres and seconds.

Boat \(A\) starts from the origin \(O\) and moves with constant speed \(5\sqrt3\text{ m s}^{-1}\) on a bearing of \(030^\circ\).

After 100 seconds boat \(B\) starts from point \(P\), which has position vector \(\begin{pmatrix}0\\1000\end{pmatrix}\).

Boat \(B\) moves with constant speed \(10\text{ m s}^{-1}\) on a bearing of \(060^\circ\).

1. Find the velocity of each boat in vector form.[2]
2. Show that the two boats will collide.[5]