| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring standard techniques: converting complex numbers to exponential form (finding modulus and argument) and using De Moivre's theorem. Part (i) involves routine multiplication by conjugate or direct conversion, and part (ii) is a simple application of the result. While it's Further Maths content, it requires no novel insight—just methodical application of well-practiced techniques. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
2 (i) Express $\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$.\\
(ii) Hence find the smallest positive value of $n$ for which $\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }$ is real and positive.
\hfill \mbox{\textit{OCR FP3 2009 Q2 [5]}}