7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8bd0bc33-e69e-4e51-aae7-288810c5db07-6_643_1374_173_351}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows a sketch of the curve $C _ { 1 }$ with equation $y = \mathrm { f } ( x )$ where

$$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$

The lines $x = 0$ and $y = \frac { x } { 3 }$ are asymptotes to $C _ { 1 }$. The point $A$ on $C _ { 1 }$ is a minimum and the point $B$ on $C _ { 1 }$ is a maximum.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $A$ and $B$.

There is a normal to $C _ { 1 }$, which does not intersect $C _ { 1 }$ a second time, that has equation $x = k$, where $k > 0$.
\item Write down the value of $k$.

The point $P ( \alpha , \beta ) , \alpha > 0$ and $\alpha \neq k$, lies on $C _ { 1 }$. The normal to $C _ { 1 }$ at $P$ does not intersect $C _ { 1 }$ a second time.
\item Find the value of $\alpha$, leaving your answer in simplified surd form.
\item Find the equation of this normal.

The curve $C _ { 2 }$ has equation $y = | \mathrm { f } ( x ) |$
\item Sketch $C _ { 2 }$ stating the coordinates of any turning points and the equations of any asymptotes.

The line with equation $y = m x + 1$ does not touch or intersect $C _ { 2 }$.
\item Find the set of possible values for $m$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2013 Q7 [22]}}