3 A particle moves in a horizontal plane under the action of a single force, $\mathbf { F }$ newtons. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively. At time $t$ seconds, the velocity of the particle, $\mathbf { v } \mathrm { ms } ^ { - 1 }$, is given by

$$\mathbf { v } = 4 \mathrm { e } ^ { - 2 t } \mathbf { i } + \left( 6 t - 3 t ^ { 2 } \right) \mathbf { j }$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for the acceleration of the particle at time $t$.
\item The mass of the particle is 5 kg .
\begin{enumerate}[label=(\roman*)]
\item Find an expression for the force $\mathbf { F }$ acting on the particle at time $t$.
\item Find the magnitude of $\mathbf { F }$ when $t = 0$.
\end{enumerate}\item Find the value of $t$ when $\mathbf { F }$ acts due west.
\item When $t = 0$, the particle is at the point with position vector $( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }$.

Find the position vector, $\mathbf { r }$ metres, of the particle at time $t$.
\end{enumerate}

\hfill \mbox{\textit{AQA M2 2011 Q3 [14]}}