Two ships $P$ and $Q$ are travelling at night with constant velocities. At midnight, $P$ is at the point with position vector $(20\mathbf{i} + 10\mathbf{j})$ km relative to a fixed origin $O$. At the same time, $Q$ is at the point with position vector $(14\mathbf{i} - 6\mathbf{j})$ km. Three hours later, $P$ is at the point with position vector $(29\mathbf{i} + 34\mathbf{j})$ km. The ship $Q$ travels with velocity $12\mathbf{j}$ km h$^{-1}$. At time $t$ hours after midnight, the position vectors of $P$ and $Q$ are $\mathbf{p}$ km and $\mathbf{q}$ km respectively. Find

\begin{enumerate}[label=(\alph*)]
\item the velocity of $P$, in terms of $\mathbf{i}$ and $\mathbf{j}$, [2]
\item expressions for $\mathbf{p}$ and $\mathbf{q}$, in terms of $t$, $\mathbf{i}$ and $\mathbf{j}$. [4]
\end{enumerate}

At time $t$ hours after midnight, the distance between $P$ and $Q$ is $d$ km.

\begin{enumerate}[label=(\alph*)]\setcounter{enumi}{2}
\item By finding an expression for $\overrightarrow{PQ}$, show that
$$d^2 = 25t^2 - 92t + 292.$$ [5]
\end{enumerate}

Weather conditions are such that an observer on $P$ can only see the lights on $Q$ when the distance between $P$ and $Q$ is 15 km or less. Given that when $t = 1$, the lights on $Q$ move into sight of the observer,

\begin{enumerate}[label=(\alph*)]\setcounter{enumi}{3}
\item find the time, to the nearest minute, at which the lights on $Q$ move out of sight of the observer. [5]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M1 2005 Q7 [16]}}