| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Challenging +1.8 This question requires finding stationary points by differentiating a product (exponential and trig), solving f'(x)=0 to identify the arithmetic pattern in x-values, then substituting back to show y-values form a GP. Part (b) requires conceptual understanding of periodicity and exponential decay. While multi-step and requiring insight into sequence patterns, the calculus itself is standard A-level and the patterns emerge naturally from the structure of the function. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic1.04i Geometric sequences: nth term and finite series sum1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f'(x) = e^{-x}\cos x - e^{-x}\sin x\) | M1, A1 | Product rule; correct |
| \(f'(x) = 0\) and \(e^{-x} \neq 0 \Rightarrow \cos x = \sin x\) | E1 | AO 2.2a |
| \(\Rightarrow \tan x = 1\) | M1 | Use of \(\frac{\sin}{\cos} = \tan\) |
| \(\Rightarrow x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \frac{13\pi}{4}\) | A1 | \(x = \frac{\pi}{4}\) (condone 45°); \(\frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \ldots\) |
| So an AP with \(d = \pi\) | E1FT | Must state the common difference; FT their values of \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = \frac{\sqrt{2}}{2}e^{-\frac{\pi}{4}}, -\frac{\sqrt{2}}{2}e^{-\frac{5\pi}{4}}, \frac{\sqrt{2}}{2}e^{-\frac{9\pi}{4}}, -\frac{\sqrt{2}}{2}e^{-\frac{13\pi}{4}}\) | M1, A1 | Substituting one value of \(x\) into \(f(x)\); correct |
| This is a GP with \(r = -e^{-\pi}\) | E1FT | Must state common ratio; FT their values of \(y\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Yes with explanation that values of \(x\) would continue to be separated by \(\pi\) and so values of \(y\) would continue to have same common ratio | E1 | AO 2.2a |
# Question 11:
## Part (a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = e^{-x}\cos x - e^{-x}\sin x$ | M1, A1 | Product rule; correct |
| $f'(x) = 0$ and $e^{-x} \neq 0 \Rightarrow \cos x = \sin x$ | E1 | AO 2.2a |
| $\Rightarrow \tan x = 1$ | M1 | Use of $\frac{\sin}{\cos} = \tan$ |
| $\Rightarrow x = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \frac{13\pi}{4}$ | A1 | $x = \frac{\pi}{4}$ (condone 45°); $\frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \ldots$ |
| So an AP with $d = \pi$ | E1FT | Must state the common difference; FT their values of $x$ |
## Part (a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = \frac{\sqrt{2}}{2}e^{-\frac{\pi}{4}}, -\frac{\sqrt{2}}{2}e^{-\frac{5\pi}{4}}, \frac{\sqrt{2}}{2}e^{-\frac{9\pi}{4}}, -\frac{\sqrt{2}}{2}e^{-\frac{13\pi}{4}}$ | M1, A1 | Substituting one value of $x$ into $f(x)$; correct |
| This is a GP with $r = -e^{-\pi}$ | E1FT | Must state common ratio; FT their values of $y$ |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Yes with explanation that values of $x$ would continue to be separated by $\pi$ and so values of $y$ would continue to have same common ratio | E1 | AO 2.2a |
---11 The curve $y = \mathrm { f } ( x )$ is defined by the function $\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x$ with domain $0 \leq x \leq 4 \pi$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of the stationary points of the curve $y = \mathrm { f } ( x )$, when arranged in increasing order, form an arithmetic sequence.
\item Show that the corresponding $y$-coordinates form a geometric sequence.
\end{enumerate}\item Would the result still hold with a larger domain? Give reasons for your answer.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 Q11 [10]}}