| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question requiring standard techniques: converting to exponential form (either by rationalizing or converting numerator/denominator separately) and using De Moivre's theorem to find when the argument becomes a multiple of 2π. The calculations are routine with no conceptual surprises, making it slightly easier than average even for FP3. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02f Convert between forms: cartesian and modulus-argument |
\begin{enumerate}[label=(\roman*)]
\item Express $\frac{\sqrt{3} + i}{\sqrt{3} - i}$ in the form $re^{i\theta}$, where $r > 0$ and $0 \leqslant \theta < 2\pi$. [3]
\item Hence find the smallest positive value of $n$ for which $\left(\frac{\sqrt{3} + i}{\sqrt{3} - i}\right)^n$ is real and positive. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q2 [5]}}