| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Groups |
| Type | Verify group axioms |
| Difficulty | Standard +0.8 This is a Further Maths group theory question requiring understanding of group axioms and complex number multiplication. Part (i) is routine inverse calculation, but part (ii) requires proving closure with careful attention to the restricted domain for θ, and part (iii) tests understanding of how the domain restriction affects powers. The domain constraint makes this more subtle than standard complex number manipulation. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02f Convert between forms: cartesian and modulus-argument4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)8.03c Group definition: recall and use, show structure is/isn't a group8.03d Latin square property: for group tables |
2 It is given that the set of complex numbers of the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ for $- \pi < \theta \leqslant \pi$ and $r > 0$, under multiplication, forms a group.\\
(i) Write down the inverse of $5 \mathrm { e } ^ { \frac { 1 } { 3 } \pi \mathrm { i } }$.\\
(ii) Prove the closure property for the group.\\
(iii) $Z$ denotes the element $\mathrm { e } ^ { \mathrm { i } \gamma }$, where $\frac { 1 } { 2 } \pi < \gamma < \pi$. Express $Z ^ { 2 }$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$, where $- \pi < \theta < 0$.
\hfill \mbox{\textit{OCR FP3 2009 Q2 [5]}}