| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - polynomial/exponential products |
| Difficulty | Standard +0.3 This question requires finding a second derivative to verify an inflection point (standard product rule application twice) and computing an area using integration by parts. Both are routine A-level techniques with no novel insight required. The 'show that' format provides target answers to work towards, making it slightly easier than average. |
| Spec | 1.07p Points of inflection: using second derivative1.08e Area between curve and x-axis: using definite integrals |
\includegraphics{figure_8}
The diagram shows the curve $M$ with equation $y = xe^{-2x}$.
\begin{enumerate}[label=(\alph*)]
\item Show that $M$ has a point of inflection at the point $P$ where $x = 1$. [5]
\end{enumerate}
The line $L$ passes through the origin $O$ and the point $P$. The shaded region $R$ is enclosed by the curve $M$ and the line $L$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the area of $R$ is given by
$$\frac{1}{4}(a + be^{-2}),$$
where $a$ and $b$ are integers to be determined. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2021 Q8 [11]}}