| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative satisfies condition |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C3 techniques: odd/even function properties, quotient rule differentiation, substitution integration, and function transformations. While it requires multiple skills (differentiation, integration, sketching), each part follows routine procedures with clear guidance. The substitution is explicitly suggested, and the final part simply requires recognizing the relationship between transformations and definite integrals. Slightly easier than average due to the scaffolding and standard methods throughout. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07q Product and quotient rules: differentiation1.08h Integration by substitution |
The function $\text{f}(x) = \frac{\sin x}{2 - \cos x}$ has domain $-\pi \leqslant x \leqslant \pi$.
Fig. 8 shows the graph of $y = \text{f}(x)$ for $0 \leqslant x \leqslant \pi$.
\includegraphics{figure_6}
\begin{enumerate}[label=(\roman*)]
\item Find $\text{f}(-x)$ in terms of $\text{f}(x)$. Hence sketch the graph of $y = \text{f}(x)$ for the complete domain $-\pi \leqslant x \leqslant \pi$. [3]
\item Show that $\text{f}'(x) = \frac{2\cos x - 1}{(2 - \cos x)^2}$. Hence find the exact coordinates of the turning point P.
State the range of the function $\text{f}(x)$, giving your answer exactly. [8]
\item Using the substitution $u = 2 - \cos x$ or otherwise, find the exact value of $\int_0^\pi \frac{\sin x}{2 - \cos x} dx$. [4]
\item Sketch the graph of $y = \text{f}(2x)$. [1]
\item Using your answers to parts (iii) and (iv), write down the exact value of $\int_0^{\frac{\pi}{2}} \frac{\sin 2x}{2 - \cos 2x} dx$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q6 [18]}}