| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Sum geometric series with complex terms |
| Difficulty | Challenging +1.3 This is a standard FP3 question on summing trigonometric series using complex exponentials and geometric series. Part (i) requires recognizing the geometric series formula with complex terms, part (ii) involves routine manipulation of exponentials to trigonometric form, and part (iii) is straightforward equation solving. While it requires multiple techniques and careful algebra, these are well-practiced methods in Further Maths with no novel insight needed. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.06d Natural logarithm: ln(x) function and properties4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(C + i S = 1 + e^{i\theta} + e^{2i\theta} + e^{3i\theta} + e^{4i\theta} + e^{5i\theta}\); \(= \frac{e^{6i\theta}-1}{e^{i\theta}-1}\); \(= \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta/2} - e^{-i\theta/2}} \cdot \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta/2} - e^{-i\theta/2}}\) | M1; M1; A1; A1 4 | For using de Moivre, showing at least 3 terms; For recognising GP; For correct GP sum; For obtaining correct expression AG |
| (ii) \(C + i S = \frac{2i\sin 3\theta}{2i\sin\frac{1}{2}\theta} \cdot e^{5i\theta}\); Re \(\Rightarrow C = \sin 30 \cos\frac{5}{2}\theta \cosec\frac{1}{2}\theta\); Im \(\Rightarrow S = \sin 30\sin\frac{5}{2}\theta \cosec\frac{1}{2}\theta\) | M1; A1; A1; B1 4 | For expressing numerator and denominator in terms of sines; For \(k\sin 30\) and \(k\sin\frac{1}{2}\theta\); For correct expression AG; For correct expression |
| (iii) \(C = S \Rightarrow \sin 30 = 0, \tan\frac{5}{2}\theta = 1\); \(\theta = \frac{1}{3}\pi, \frac{2}{3}\pi\); \(\theta = \frac{1}{10}\pi, \frac{1}{2}\pi, \frac{9}{10}\pi\) | M1; A1; A2 4 [12] | For either equation deduced AEF. Ignore values outside \(0 < \theta < \pi\); For both values correct and no extras. Allow A1 for any 1 value OR all correct with extras |
**(i)** $C + i S = 1 + e^{i\theta} + e^{2i\theta} + e^{3i\theta} + e^{4i\theta} + e^{5i\theta}$; $= \frac{e^{6i\theta}-1}{e^{i\theta}-1}$; $= \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta/2} - e^{-i\theta/2}} \cdot \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta/2} - e^{-i\theta/2}}$ | M1; M1; A1; A1 4 | For using de Moivre, showing at least 3 terms; For recognising GP; For correct GP sum; For obtaining correct expression **AG**
**(ii)** $C + i S = \frac{2i\sin 3\theta}{2i\sin\frac{1}{2}\theta} \cdot e^{5i\theta}$; Re $\Rightarrow C = \sin 30 \cos\frac{5}{2}\theta \cosec\frac{1}{2}\theta$; Im $\Rightarrow S = \sin 30\sin\frac{5}{2}\theta \cosec\frac{1}{2}\theta$ | M1; A1; A1; B1 4 | For expressing numerator and denominator in terms of sines; For $k\sin 30$ and $k\sin\frac{1}{2}\theta$; For correct expression **AG**; For correct expression
**(iii)** $C = S \Rightarrow \sin 30 = 0, \tan\frac{5}{2}\theta = 1$; $\theta = \frac{1}{3}\pi, \frac{2}{3}\pi$; $\theta = \frac{1}{10}\pi, \frac{1}{2}\pi, \frac{9}{10}\pi$ | M1; A1; A2 4 [12] | For either equation deduced **AEF**. Ignore values outside $0 < \theta < \pi$; For both values correct and no extras. Allow A1 for any 1 value **OR** all correct with extras
---The series $C$ and $S$ are defined for $0 < \theta < \pi$ by
\begin{align}
C &= 1 + \cos \theta + \cos 2\theta + \cos 3\theta + \cos 4\theta + \cos 5\theta, \\
S &= \sin \theta + \sin 2\theta + \sin 3\theta + \sin 4\theta + \sin 5\theta.
\end{align}
\begin{enumerate}[label=(\roman*)]
\item Show that $C + iS = \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta} - e^{-i\theta}} \cdot e^{i\theta}$. [4]
\item Deduce that $C = \sin 3\theta \cos \frac{5}{2}\theta \operatorname{cosec} \frac{1}{2}\theta$ and write down the corresponding expression for $S$. [4]
\item Hence find the values of $\theta$, in the range $0 < \theta < \pi$, for which $C = S$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2006 Q7 [12]}}