[In this question $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin $O$.]

Two ships $P$ and $Q$ are moving with constant velocities. Ship $P$ moves with velocity $(2\mathbf{i} - 3\mathbf{j})$ km h$^{-1}$ and ship $Q$ moves with velocity $(3\mathbf{i} + 4\mathbf{j})$ km h$^{-1}$.

\begin{enumerate}[label=(\alph*)]
\item Find, to the nearest degree, the bearing on which $Q$ is moving. [2]
\end{enumerate}

At 2 pm, ship $P$ is at the point with position vector $(\mathbf{i} + \mathbf{j})$ km and ship $Q$ is at the point with position vector $(-2\mathbf{j})$ km.

At time $t$ hours after 2 pm, the position vector of $P$ is $\mathbf{p}$ km and the position vector of $Q$ is $\mathbf{q}$ km.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Write down expressions, in terms of $t$, for
\begin{enumerate}[label=(\roman*)]
\item $\mathbf{p}$,
\item $\mathbf{q}$,
\item $\overrightarrow{PQ}$. [5]
\end{enumerate}

\item Find the time when
\begin{enumerate}[label=(\roman*)]
\item $Q$ is due north of $P$,
\item $Q$ is north-west of $P$. [4]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2011 Q7 [11]}}