| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2017 |
| Session | Specimen |
| Marks | 10 |
| Topic | Complex numbers 2 |
| Type | Basic roots of unity properties |
| Difficulty | Standard +0.3 This is a structured question on basic roots of unity properties with clear guidance. Part (i) asks students to justify standard facts about fifth roots (equal moduli, equal angular spacing, conjugate pairs, sum equals zero) - all straightforward applications of theory. Part (ii) requires finding midpoints and forming a polynomial, which involves routine algebraic manipulation. While it covers multiple concepts, each step is scaffolded and uses well-known properties, making it slightly easier than average. |
| Spec | 4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation4.02r nth roots: of complex numbers |
10 The Argand diagram below shows the origin $O$ and pentagon $A B C D E$, where $A , B , C , D$ and $E$ are the points that represent the complex numbers $a , b , c , d$ and $e$, and where $a$ is a positive real number. You are given that these five complex numbers are the roots of the equation $z ^ { 5 } - a ^ { 5 } = 0$.\\
\includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}\\
(i) Justify each of the following statements.
\begin{enumerate}[label=(\alph*)]
\item $A , B , C , D$ and $E$ lie on a circle with centre $O$.
\item $A B C D E$ is a regular pentagon.
\item $b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c$
\item $b ^ { * } = e$
\item $a + b + c + d + e = 0$\\
(ii) The midpoints of sides $A B , B C , C D , D E$ and $E A$ represent the complex numbers $p , q , r , s$ and $t$. Determine a polynomial equation, with real coefficients, that has roots $p , q , r , s$ and $t$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2017 Q10 [10]}}