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

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

A particle $P$ is moving on a smooth horizontal plane.\\
At time $t$ seconds $( t \geqslant 0 )$, the position vector of $P$, relative to a fixed point $O$, is $\mathbf { r }$ metres and the velocity of $P$ is $\mathbf { v } \mathrm { ms } ^ { - 1 }$ where

$$\mathbf { v } = \left( 4 t ^ { 2 } - 5 t \right) \mathbf { i } + ( - 10 t - 12 ) \mathbf { j }$$

When $t = 0 , \mathbf { r } = 2 \mathbf { i } + 6 \mathbf { j }$\\
(a) Find $\mathbf { r }$ when $t = 2$

When $t = T$ particle $P$ is moving in the direction of the vector $\mathbf { i } - 2 \mathbf { j }$\\
(b) Find the value of $T$\\
(c) Find the exact magnitude of the acceleration of $P$ when $t = 2.5$

\hfill \mbox{\textit{Edexcel M2 2023 Q2 [10]}}