[In this question, the unit vectors $\mathbf{i}$ and $\mathbf{j}$ are in a vertical plane, $\mathbf{i}$ being horizontal and $\mathbf{j}$ being vertically upwards.]

At time $t = 0$, a particle $P$ is projected with velocity $(4\mathbf{i} + 9\mathbf{j})$ m s$^{-1}$ from a fixed point $O$ on horizontal ground. The particle moves freely under gravity. When $P$ is at the point $H$ on its path, $P$ is at its greatest height above the ground.

\begin{enumerate}[label=(\alph*)]
\item Find the time taken by $P$ to reach $H$. [2]
\end{enumerate}

At the point $A$ on its path, the position vector of $P$ relative to $O$ is $(k\mathbf{i} + k\mathbf{j})$ m, where $k$ is a positive constant.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $k$. [4]

\item Find, in terms of $k$, the position vector of the other point on the path of $P$ which is at the same vertical height above the ground as the point $A$. [3]
\end{enumerate}

At time $T$ seconds the particle is at the point $B$ and is moving perpendicular to $(4\mathbf{i} + 9\mathbf{j})$

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the value of $T$. [4]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2 2015 Q7 [13]}}