| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | De Moivre to derive trigonometric identities |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 question testing routine De Moivre's theorem applications. Part (a) requires expanding brackets and using standard identities (z^n + 1/z^n = 2cos(nθ)), while part (b) involves straightforward nth roots and modulus-argument form. The algebraic manipulation in (a)(ii) requires care but follows a well-practiced technique. More challenging than typical A-level Core questions due to the Further Maths content, but these are textbook exercises without requiring novel insight. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z^n + \frac{1}{z^n} = 2\cos n\theta\) | B1 | |
| \(z^n - \frac{1}{z^n} = 2j\sin n\theta\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((z-\frac{1}{z})^4 = (2j\sin\theta)^4 = 16\sin^4\theta\) | M1 | |
| \((z+\frac{1}{z})^2 = (2\cos\theta)^2 = 4\cos^2\theta\) | M1 | |
| Expanding \((z-\frac{1}{z})^4(z+\frac{1}{z})^2\) and collecting terms | M1 A1 | |
| \(A=\frac{1}{16}, B=-\frac{1}{4}, C=\frac{3}{8}, D=\frac{1}{4}\) (or equivalent) | A1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | 4+4j | = 4\sqrt{2}\) |
| \(\arg(4+4j) = \frac{\pi}{4}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r = (4\sqrt{2})^{1/5} = 2^{3/10}\cdot 2^{1/10} = 2^{2/5}\cdot ... = \sqrt[10]{32\sqrt{2}}\) | M1 A1 | |
| \(\theta = \frac{\pi/4 + 2k\pi}{5}\) for \(k = 0, \pm1, \pm2\) | M1 A1 | |
| Five roots correctly stated | A1 | |
| Argand diagram with 5 equally spaced points on circle | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One of the fifth roots must be real and positive integer form | M1 | |
| \(p=1, q=1\) (since \((1+j)^5 = 4+4j\)) | A1 |
# Question 2:
## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z^n + \frac{1}{z^n} = 2\cos n\theta$ | B1 | |
| $z^n - \frac{1}{z^n} = 2j\sin n\theta$ | B1 | |
## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(z-\frac{1}{z})^4 = (2j\sin\theta)^4 = 16\sin^4\theta$ | M1 | |
| $(z+\frac{1}{z})^2 = (2\cos\theta)^2 = 4\cos^2\theta$ | M1 | |
| Expanding $(z-\frac{1}{z})^4(z+\frac{1}{z})^2$ and collecting terms | M1 A1 | |
| $A=\frac{1}{16}, B=-\frac{1}{4}, C=\frac{3}{8}, D=\frac{1}{4}$ (or equivalent) | A1 A1 | |
## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|4+4j| = 4\sqrt{2}$ | B1 | |
| $\arg(4+4j) = \frac{\pi}{4}$ | B1 | |
## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r = (4\sqrt{2})^{1/5} = 2^{3/10}\cdot 2^{1/10} = 2^{2/5}\cdot ... = \sqrt[10]{32\sqrt{2}}$ | M1 A1 | |
| $\theta = \frac{\pi/4 + 2k\pi}{5}$ for $k = 0, \pm1, \pm2$ | M1 A1 | |
| Five roots correctly stated | A1 | |
| Argand diagram with 5 equally spaced points on circle | B1 | |
## Part (b)(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| One of the fifth roots must be real and positive integer form | M1 | |
| $p=1, q=1$ (since $(1+j)^5 = 4+4j$) | A1 | |
---2
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Given that $z = \cos \theta + \mathrm { j } \sin \theta$, express $z ^ { n } + \frac { 1 } { z ^ { n } }$ and $z ^ { n } - \frac { 1 } { z ^ { n } }$ in simplified trigonometric form.
\item By considering $\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }$, find $A , B , C$ and $D$ such that
$$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the modulus and argument of $4 + 4 \mathrm { j }$.
\item Find the fifth roots of $4 + 4 \mathrm { j }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Illustrate these fifth roots on an Argand diagram.
\item Find integers $p$ and $q$ such that $( p + q \mathrm { j } ) ^ { 5 } = 4 + 4 \mathrm { j }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2006 Q2 [18]}}