7.
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\caption{Figure 3}
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\end{figure}
The curve \(C\) with equation \(y = 2 \mathrm { e } ^ { x } + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3 .
- Find the equation of the normal to \(C\) at \(M\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N ( n , 0 )\).
- Show that \(n = 14\).
The point \(P ( \ln 4,13 )\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(P N\), as shown in Fig. 3.
- Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found.