| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Moderate -0.8 This is a straightforward application of exponential and polar form rules for complex numbers. Part (a) requires multiplying/dividing moduli and adding/subtracting arguments—direct formula application. Part (b) uses De Moivre's theorem with negative exponent. Both parts are routine FP3 exercises with no problem-solving or conceptual challenges, making this easier than average even for Further Maths. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((z_1z_2) = 6e^{i\frac{23}{12}\pi}\); \(\frac{z_1}{z_2} = \frac{2}{3}e^{-i\frac{1}{12}\pi} = \frac{2}{3}e^{i\frac{23}{12}\pi}\) | B1; B1; M1 A1 4 | For modulus = 6; For argument = \(\frac{23}{12}\pi\); For subtracting arguments; For correct answer |
| (b) \((w^{-5}) = 2^{-5}\cis(-\frac{5}{8}\pi)\) = \(\frac{1}{32}(\cos\frac{11}{8}\pi + i\sin\frac{11}{8}\pi)\) | M1; A1; A1 3 [7] | For use of de Moivre; For \(-\frac{5}{8}\pi\) seen or implied; For correct answer (allow \(2^{-5}\) and \(\cis\frac{11}{8}\pi\)) |
**(a)** $(z_1z_2) = 6e^{i\frac{23}{12}\pi}$; $\frac{z_1}{z_2} = \frac{2}{3}e^{-i\frac{1}{12}\pi} = \frac{2}{3}e^{i\frac{23}{12}\pi}$ | B1; B1; M1 A1 4 | For modulus = 6; For argument = $\frac{23}{12}\pi$; For subtracting arguments; For correct answer
**(b)** $(w^{-5}) = 2^{-5}\cis(-\frac{5}{8}\pi)$ = $\frac{1}{32}(\cos\frac{11}{8}\pi + i\sin\frac{11}{8}\pi)$ | M1; A1; A1 3 [7] | For use of de Moivre; For $-\frac{5}{8}\pi$ seen or implied; For correct answer (allow $2^{-5}$ and $\cis\frac{11}{8}\pi$)
---\begin{enumerate}[label=(\alph*)]
\item Given that $z_1 = 2e^{\frac{5\pi i}{6}}$ and $z_2 = 3e^{\frac{2\pi i}{3}}$, express $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $re^{i\theta}$, where $r > 0$ and $0 \leq \theta < 2\pi$. [4]
\item Given that $w = 2(\cos \frac{1}{3}\pi + i \sin \frac{1}{3}\pi)$, express $w^{-5}$ in the form $r(\cos \theta + i \sin \theta)$, where $r > 0$ and $0 \leq \theta < 2\pi$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2006 Q2 [7]}}