\begin{enumerate}
  \item \hspace{0pt} [In this question, the perpendicular unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in a horizontal plane.]
\end{enumerate}

A particle $Q$ of mass 1.5 kg is moving on a smooth horizontal plane under the action of a single force $\mathbf { F }$ newtons. At time $t$ seconds ( $t \geqslant 0$ ), the position vector of $Q$, relative to a fixed point $O$, is $\mathbf { r }$ metres and the velocity of $Q$ is $\mathbf { v } \mathrm { ms } ^ { - 1 }$\\
It is given that

$$\mathbf { v } = \left( 3 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t ^ { 3 } + k t \right) \mathbf { j }$$

where $k$ is a constant.\\
Given that when $t = 2$ particle $Q$ is moving in the direction of the vector $\mathbf { i } + \mathbf { j }$\\
(a) show that $k = 4$\\
(b) find the magnitude of $\mathbf { F }$ when $t = 2$

Given that $\mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j }$ when $t = 0$\\
(c) find $\mathbf { r }$ when $t = 2$

\hfill \mbox{\textit{Edexcel M2 2023 Q4 [10]}}