| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Roots of unity with derived equations |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on finding nth roots of unity and related complex numbers. Part (i) requires standard application of de Moivre's theorem to find fifth roots of unity, and part (ii) extends this by factoring out a constant. While it's Further Maths content (inherently harder than standard A-level), it's a routine textbook exercise requiring direct application of known techniques rather than problem-solving or insight, placing it slightly above average difficulty overall. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z = 1, e^{2\pi i/5}, e^{4\pi i/5}, e^{6\pi i/5}, e^{8\pi i/5}\) | M1 | \(e^{2\pi i/5}\) soi |
| A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z^5 = -32\) has root \(-2\), so roots are \(-2, -2e^{2\pi i/5}, -2e^{4\pi i/5}, -2e^{6\pi i/5}, -2e^{8\pi i/5}\) | M1 | Use part (i) or from scratch |
| Roots \(-2, 2e^{7\pi i/5}, 2e^{9\pi i/5}, 2e^{\pi i/5}, 2e^{3\pi i/5}\) | A1 | cao with \(r>0, 0<\theta<2\pi\) (allow \(2e^{\pi i}\) for \(-2\)) |
| Argand diagram | M1 | One root in each quadrant plus one on real axis |
| A1 | Axes and roots labelled. Roots equal moduli and equiangular spacing |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z = 1, e^{2\pi i/5}, e^{4\pi i/5}, e^{6\pi i/5}, e^{8\pi i/5}$ | M1 | $e^{2\pi i/5}$ soi |
| | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z^5 = -32$ has root $-2$, so roots are $-2, -2e^{2\pi i/5}, -2e^{4\pi i/5}, -2e^{6\pi i/5}, -2e^{8\pi i/5}$ | M1 | Use part (i) or from scratch |
| Roots $-2, 2e^{7\pi i/5}, 2e^{9\pi i/5}, 2e^{\pi i/5}, 2e^{3\pi i/5}$ | A1 | cao with $r>0, 0<\theta<2\pi$ (allow $2e^{\pi i}$ for $-2$) |
| Argand diagram | M1 | One root in each quadrant plus one on real axis |
| | A1 | Axes and roots labelled. Roots equal moduli and equiangular spacing |
---1 In this question, give all non-real numbers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ where $r > 0$ and $0 < \theta < 2 \pi$.\\
(i) Solve $z ^ { 5 } = 1$.\\
(ii) Hence, or otherwise, solve $z ^ { 5 } + 32 = 0$. Sketch an Argand diagram showing the roots.
\hfill \mbox{\textit{OCR FP3 2016 Q1 [6]}}