| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Basic roots of unity properties |
| Difficulty | Standard +0.8 This FP3 question requires understanding of complex numbers in polar form, geometric series, and roots of unity. Part (i) is routine application of De Moivre's theorem. Part (ii) requires recognizing that the partial sums form a geometric pattern and visualizing the spiral structure in the Argand diagram—this demands geometric insight beyond standard exercises. Part (iii) is straightforward recall that w is a 5th root of unity. The visualization and geometric reasoning elevate this above typical complex number questions but it remains within standard FP3 territory. |
| Spec | 4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(w^2 = \cos\frac{4\pi}{5} + i\sin\frac{4\pi}{5}\) | B1 | For correct value |
| \(w^3 = \cos\frac{6\pi}{5} + i\sin\frac{6\pi}{5}\) | B1 | For correct value |
| \(w^6 = \cos\frac{2\pi}{5} - i\sin\frac{2\pi}{5}\) | B1 | For \(w^*\) seen or implied |
| \(= \cos\frac{8\pi}{5} + i\sin\frac{8\pi}{5}\) | B1 4 | For correct value |
| SR For exponential form with \(i\) missing, award B0 first time, allow others |
| Answer | Marks |
|---|---|
| B1* | For \(1+w\) in approximately correct position |
| B1 (*dep) | For \(AB = BC = CD\) |
| B1 (*dep) | For \(BC\), \(CD\) equally inclined to Im axis |
| B1 4 | For \(E\) at the origin |
| Allow points joined by arcs, or not joined. Labels not essential |
| Answer | Marks | Guidance |
|---|---|---|
| \(z^5 - 1 = 0\) OR \(z^5 + z^4 + z^3 + z^2 + z = 0\) | B1 1 | For correct equation AEF (in any variable) |
| Allow factorised forms using \(w\), exp or trig |
## (i)
Allow $\cis\frac{2\pi}{5}$ and $e^{\frac{8\pi i}{5}}$ throughout
$w^2 = \cos\frac{4\pi}{5} + i\sin\frac{4\pi}{5}$ | B1 | For correct value
$w^3 = \cos\frac{6\pi}{5} + i\sin\frac{6\pi}{5}$ | B1 | For correct value
$w^6 = \cos\frac{2\pi}{5} - i\sin\frac{2\pi}{5}$ | B1 | For $w^*$ seen or implied
$= \cos\frac{8\pi}{5} + i\sin\frac{8\pi}{5}$ | B1 4 | For correct value
| | SR For exponential form with $i$ missing, award B0 first time, allow others
## (ii)
| B1* | For $1+w$ in approximately correct position
| B1 (*dep) | For $AB = BC = CD$
| B1 (*dep) | For $BC$, $CD$ equally inclined to Im axis
| B1 4 | For $E$ at the origin
| | Allow points joined by arcs, or not joined. Labels not essential
## (iii)
$z^5 - 1 = 0$ OR $z^5 + z^4 + z^3 + z^2 + z = 0$ | B1 1 | For correct equation AEF (in any variable)
| | Allow factorised forms using $w$, exp or trig
---In this question, $w$ denotes the complex number $\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$.
\begin{enumerate}[label=(\roman*)]
\item Express $w^2$, $w^3$ and $w^4$ in polar form, with arguments in the interval $0 \leq \theta < 2\pi$. [4]
\item The points in an Argand diagram which represent the numbers
$$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$
are denoted by $A$, $B$, $C$, $D$, $E$ respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
\item Write down a polynomial equation of degree 5 which is satisfied by $w$. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2010 Q3 [9]}}