9 Fig. 9 shows the curve \(y = x \mathrm { e } ^ { - 2 x }\) together with the straight line \(y = m x\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P . The dashed line is the tangent at P .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1a06289-d9e9-4f6b-ab58-70db1a4748ef-4_424_972_383_559}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{figure}
- Show that the \(x\)-coordinate of P is \(- \frac { 1 } { 2 } \ln m\).
- Find, in terms of \(m\), the gradient of the tangent to the curve at P .
You are given that OP and this tangent are equally inclined to the \(x\)-axis.
- Show that \(m = \mathrm { e } ^ { - 2 }\), and find the exact coordinates of P .
- Find the exact area of the shaded region between the line OP and the curve.
\section*{END OF QUESTION PAPER}
\section*{OCR \(^ { \text {N } }\)}