| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Sum geometric series with complex terms |
| Difficulty | Challenging +1.2 This is a structured Further Maths question requiring knowledge of complex exponentials, geometric series, and De Moivre's theorem. Part (i) is routine recall, part (ii) is straightforward algebra with conjugates, and part (iii) requires recognizing the series as Re/Im of a geometric series with complex terms—a standard FP2 technique. While it involves multiple steps and further maths content, the approach is methodical without requiring novel insight. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | w | = \frac{1}{2}\), \(\arg(w) = 3\theta\) |
| \( | w^* | = \frac{1}{2}\), \(\arg(w^*) = -3\theta\) |
| \( | jw | = \frac{1}{2}\), \(\arg(jw) = 3\theta + \frac{\pi}{2}\) |
| Argand diagram: \(w\) and \(w^*\) conjugate pair, \(jw\) correctly rotated | B1+B1+B1 | Points correctly placed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1+w)(1+w^*) = 1 + w + w^* + ww^*\) | M1 | Expanding |
| \(w + w^* = 2\text{Re}(w) = \cos 3\theta\) | A1 | Using real part |
| \(ww^* = | w | ^2 = \frac{1}{4}\) |
| \(= 1 + \cos 3\theta + \frac{1}{4} = \frac{5}{4} + \cos 3\theta\) | A1 | Completing proof |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(C + jS = e^{2j\theta} - \frac{1}{2}e^{5j\theta} + \frac{1}{4}e^{8j\theta} - \cdots\) | M1 | Forming combined series |
| \(= e^{2j\theta}\left(1 - \frac{1}{2}e^{3j\theta} + \cdots\right) = \frac{e^{2j\theta}}{1+\frac{1}{2}e^{3j\theta}}\) | M1 A1 | Summing GP with \(r = -\frac{1}{2}e^{3j\theta}\) |
| \(= \frac{2e^{2j\theta}}{2 + e^{3j\theta}}\) | A1 | |
| Multiply numerator and denominator by conjugate \(2 + e^{-3j\theta}\) | M1 | |
| Denominator: \(4 + 2\cos 3\theta + 2\cos 3\theta + 1 = 5 + 4\cos 3\theta\) | A1 | |
| Numerator real part: \(2(\cos 2\theta + \frac{1}{2}\cos(-\theta) + \frac{1}{2}\cos 5\theta)\)... gives \(4\cos 2\theta + 2\cos\theta\) | M1 A1 | |
| \(C = \frac{4\cos 2\theta + 2\cos\theta}{5 + 4\cos 3\theta}\) | A1 | As required |
| \(S = \frac{4\sin 2\theta - 2\sin\theta}{5 + 4\cos 3\theta}\) | B1 | From imaginary part |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|w| = \frac{1}{2}$, $\arg(w) = 3\theta$ | B1 | |
| $|w^*| = \frac{1}{2}$, $\arg(w^*) = -3\theta$ | B1 | |
| $|jw| = \frac{1}{2}$, $\arg(jw) = 3\theta + \frac{\pi}{2}$ | B1 | |
| Argand diagram: $w$ and $w^*$ conjugate pair, $jw$ correctly rotated | B1+B1+B1 | Points correctly placed |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1+w)(1+w^*) = 1 + w + w^* + ww^*$ | M1 | Expanding |
| $w + w^* = 2\text{Re}(w) = \cos 3\theta$ | A1 | Using real part |
| $ww^* = |w|^2 = \frac{1}{4}$ | A1 | |
| $= 1 + \cos 3\theta + \frac{1}{4} = \frac{5}{4} + \cos 3\theta$ | A1 | Completing proof |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $C + jS = e^{2j\theta} - \frac{1}{2}e^{5j\theta} + \frac{1}{4}e^{8j\theta} - \cdots$ | M1 | Forming combined series |
| $= e^{2j\theta}\left(1 - \frac{1}{2}e^{3j\theta} + \cdots\right) = \frac{e^{2j\theta}}{1+\frac{1}{2}e^{3j\theta}}$ | M1 A1 | Summing GP with $r = -\frac{1}{2}e^{3j\theta}$ |
| $= \frac{2e^{2j\theta}}{2 + e^{3j\theta}}$ | A1 | |
| Multiply numerator and denominator by conjugate $2 + e^{-3j\theta}$ | M1 | |
| Denominator: $4 + 2\cos 3\theta + 2\cos 3\theta + 1 = 5 + 4\cos 3\theta$ | A1 | |
| Numerator real part: $2(\cos 2\theta + \frac{1}{2}\cos(-\theta) + \frac{1}{2}\cos 5\theta)$... gives $4\cos 2\theta + 2\cos\theta$ | M1 A1 | |
| $C = \frac{4\cos 2\theta + 2\cos\theta}{5 + 4\cos 3\theta}$ | A1 | As required |
| $S = \frac{4\sin 2\theta - 2\sin\theta}{5 + 4\cos 3\theta}$ | B1 | From imaginary part |
---2 In this question, $\theta$ is a real number with $0 < \theta < \frac { 1 } { 6 } \pi$, and $w = \frac { 1 } { 2 } \mathrm { e } ^ { 3 \mathrm { j } \theta }$.\\
(i) State the modulus and argument of each of the complex numbers
$$w , \quad w ^ { * } \quad \text { and } \quad \mathrm { j } w .$$
Illustrate these three complex numbers on an Argand diagram.\\
(ii) Show that $( 1 + w ) \left( 1 + w ^ { * } \right) = \frac { 5 } { 4 } + \cos 3 \theta$.
Infinite series $C$ and $S$ are defined by
$$\begin{aligned}
& C = \cos 2 \theta - \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 8 \theta - \frac { 1 } { 8 } \cos 11 \theta + \ldots \\
& S = \sin 2 \theta - \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 8 \theta - \frac { 1 } { 8 } \sin 11 \theta + \ldots
\end{aligned}$$
(iii) Show that $C = \frac { 4 \cos 2 \theta + 2 \cos \theta } { 5 + 4 \cos 3 \theta }$, and find a similar expression for $S$.
\hfill \mbox{\textit{OCR MEI FP2 2006 Q2 [18]}}