| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find gradient at specific point |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring differentiation of exponential functions, solving transcendental equations, interpreting geometric conditions (equal inclination to x-axis), and integration by parts. While individual techniques are standard C3 content, the geometric insight in part (iii) and the extended integration in part (iv) elevate this above routine exercises, though it remains within expected A-level scope with clear scaffolding. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.08e Area between curve and x-axis: using definite integrals |
Fig. 9 shows the curve $y = xe^{-2x}$ together with the straight line $y = mx$, 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.
\includegraphics{figure_9}
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinate of P is $-\frac{1}{2}\ln m$. [3]
\item Find, in terms of $m$, the gradient of the tangent to the curve at P. [4]
\end{enumerate}
You are given that OP and this tangent are equally inclined to the $x$-axis.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Show that $m = e^{-2}$, and find the exact coordinates of P. [4]
\item Find the exact area of the shaded region between the line OP and the curve. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2014 Q9 [18]}}