| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Composite transformation sketch |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring function transformations, implicit differentiation of a product involving exponential and trig functions, and a sophisticated proof involving substitution and functional relationships. Part (ii) requires differentiating f(x) = e^{-x/5}sin(x) using the product rule, setting equal to zero, and manipulating to get tan(x) = 5. Part (iii) demands algebraic manipulation to prove a functional equation, then applying u-substitution to relate two integrals—this requires careful symbolic reasoning beyond routine integration. The graphical interpretation adds another layer. While the techniques are C3-level, the question requires sustained multi-step reasoning and algebraic fluency, making it moderately challenging but not exceptional for A-level. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07j Differentiate exponentials: e^(kx) and a^(kx)1.07q Product and quotient rules: differentiation1.08d Evaluate definite integrals: between limits |
Fig. 8 shows part of the curve $y = \text{f}(x)$, where $\text{f}(x) = e^{-\frac{1}{5}x} \sin x$, for all $x$.
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\item Sketch the graphs of
(A) $y = \text{f}(2x)$,
(B) $y = \text{f}(x + \pi)$. [4]
\item Show that the $x$-coordinate of the turning point P satisfies the equation $\tan x = 5$.
Hence find the coordinates of P. [6]
\item Show that $\text{f}(x + \pi) = -e^{-\frac{1}{5}\pi}\text{f}(x)$. Hence, using the substitution $u = x - \pi$, show that
$$\int_{\pi}^{2\pi} \text{f}(x)\,dx = -e^{-\frac{1}{5}\pi} \int_{0}^{\pi} \text{f}(u)\,du.$$
Interpret this result graphically. [You should not attempt to integrate f(x).] [8]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q7 [18]}}