| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points coordinates |
| Difficulty | Standard +0.8 This question requires finding stationary points using the product rule, then proving a geometric relationship. Students must differentiate e^(-x)sin(x), solve the transcendental equation for the maximum, identify where curves touch (sin x = 1), and verify the distance relationship. The multi-step nature, combination of calculus and geometric reasoning, and need to work with transcendental equations elevates this above routine stationary point questions. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = e^{-x}\sin x \Rightarrow \frac{dy}{dx} = -e^{-x}\sin x + e^{-x}\cos x\) | M1, A1 | Product rule |
| \(= 0\) when \(\sin x = \cos x \Rightarrow x = \frac{\pi}{4}\) (in this range) | M1 | Set \(= 0\) |
| \(\Rightarrow OA' = \frac{\pi}{4}\) | A1 | |
| For intersection: \(e^{-x}\sin x = e^{-x} \Rightarrow \sin x = 1\) | M1 | Solving |
| \(\Rightarrow x = \frac{\pi}{2}\) (in this range) | A1 | |
| \(\Rightarrow OB' = \frac{\pi}{2} \Rightarrow A'B' = \frac{\pi}{4}\) | A1 | |
| Total: 7 |
## Question 7:
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = e^{-x}\sin x \Rightarrow \frac{dy}{dx} = -e^{-x}\sin x + e^{-x}\cos x$ | M1, A1 | Product rule |
| $= 0$ when $\sin x = \cos x \Rightarrow x = \frac{\pi}{4}$ (in this range) | M1 | Set $= 0$ |
| $\Rightarrow OA' = \frac{\pi}{4}$ | A1 | |
| For intersection: $e^{-x}\sin x = e^{-x} \Rightarrow \sin x = 1$ | M1 | Solving |
| $\Rightarrow x = \frac{\pi}{2}$ (in this range) | A1 | |
| $\Rightarrow OB' = \frac{\pi}{2} \Rightarrow A'B' = \frac{\pi}{4}$ | A1 | |
| | **Total: 7** | |
---7 Fig. 7 shows the graphs of the curves $y = \mathrm { e } ^ { - x }$ and $y = \mathrm { e } ^ { - x } \sin x$ for $0 \leq x \leq \pi$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-3_407_793_1085_740}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
The maximum point on $y = \mathrm { e } ^ { - x } \sin x$ is at A , and the curves touch at B .\\
$\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the points on the $x$-axis such that $\mathrm { A } ^ { \prime } \mathrm { A }$ and $\mathrm { B } ^ { \prime } \mathrm { B }$ are parallel to the $y$-axis.\\
Show that $\mathrm { OA } ^ { \prime } = \mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }$.
\hfill \mbox{\textit{OCR MEI C3 Q7 [7]}}