| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 11 |
| 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 standard FP3 question on cube roots of unity with well-established techniques. Parts (i)-(iii) involve routine manipulations using the fundamental property 1+ω+ω²=0, which students practice extensively. Part (iv) requires forming a cubic from roots using Vieta's formulas, a standard procedure. While it requires careful algebra across multiple steps (11 marks total), it follows predictable patterns without requiring novel insight or particularly challenging problem-solving. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers4.05a Roots and coefficients: symmetric functions |
The roots of the equation $z^3 - 1 = 0$ are denoted by $1, \omega$ and $\omega^2$.
\begin{enumerate}[label=(\roman*)]
\item Sketch an Argand diagram to show these roots. [1]
\item Show that $1 + \omega + \omega^2 = 0$. [2]
\item Hence evaluate
\begin{enumerate}[label=(\alph*)]
\item $(2 + \omega)(2 + \omega^2)$, [2]
\item $\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}$. [2]
\end{enumerate}
\item Hence find a cubic equation, with integer coefficients, which has roots $2, \frac{1}{2 + \omega}$ and $\frac{1}{2 + \omega^2}$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q7 [11]}}