7. (a) Find the set of values of $k$ for which the equation

$$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$

has no real roots.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-5_718_869_511_603}
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\caption{Figure 3}
\end{center}
\end{figure}

Figure 3 shows a sketch of the curve $C _ { 1 }$ with equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }$ The curve has asymptotes $x = a , x = b$ and $y = c$, where $a , b$ and $c$ are integers.\\
(b) Find the value of $a$, the value of $b$ and the value of $c$.\\
(c) Find the coordinates of the points of intersection of $C _ { 1 }$ with the line $y = 2$\\
(d) Find all the integer pairs $( r , s )$ that satisfy $s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }$

The curve $C _ { 2 }$ has equation $y = \mathrm { g } ( x )$ where $\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }$\\
(e) Show that, for suitable integers $m$ and $n , \mathrm {~g} ( x )$ can be written in the form $\mathrm { f } ( x + m ) + n$.\\
(f) Sketch $C _ { 2 }$ showing any asymptotes and stating their equations.

\hfill \mbox{\textit{Edexcel AEA 2016 Q7 [24]}}