| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Standard +0.3 This is a straightforward group theory question testing basic properties of complex numbers in polar form. Part (i) requires simple reciprocal and angle negation, part (ii) is a routine verification of closure under multiplication (multiply two elements and check the result stays in the set), and part (iii) involves doubling an angle and adjusting to the principal range. All parts are standard bookwork with no novel insight required, making it slightly easier than average for FP3. |
| Spec | 4.02d Exponential form: re^(i*theta)8.03c Group definition: recall and use, show structure is/isn't a group |
It is given that the set of complex numbers of the form $re^{i\theta}$ for $-\pi < \theta \leqslant \pi$ and $r > 0$, under multiplication, forms a group.
\begin{enumerate}[label=(\roman*)]
\item Write down the inverse of $5e^{3\pi i}$. [1]
\item Prove the closure property for the group. [2]
\item $Z$ denotes the element $e^{i\gamma}$, where $\frac{1}{2}\pi < \gamma < \pi$. Express $Z^2$ in the form $e^{i\theta}$, where $-\pi < \theta \leqslant 0$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q2 [5]}}