| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Find conjugate roots from polynomial |
| Difficulty | Moderate -0.3 This is a standard Further Maths question testing the conjugate root theorem for polynomials with real coefficients. Part (a) is immediate recall, part (b) requires forming a quadratic factor and polynomial division (routine technique), and part (c) is straightforward plotting. While it's Further Maths content, it follows a completely standard template with no problem-solving insight required, making it slightly easier than average overall. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
| VIAV SIHI NI BIIIM ION OC | VGHV SIHI NI GHIYM ION OC | VJ4V SIHI NI JIIYM ION OC |
8.
$$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$
where $a$ and $b$ are real constants.\\
Given that $- 3 + 8 \mathrm { i }$ is a complex root of the equation $\mathrm { f } ( z ) = 0$
\begin{enumerate}[label=(\alph*)]
\item write down another complex root of this equation.
\item Hence, or otherwise, find the other roots of the equation $\mathrm { f } ( z ) = 0$
\item Show on a single Argand diagram all four roots of the equation $\mathrm { f } ( z ) = 0$
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
VIAV SIHI NI BIIIM ION OC & VGHV SIHI NI GHIYM ION OC & VJ4V SIHI NI JIIYM ION OC \\
\hline
\end{tabular}
\end{center}
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2018 Q8 [9]}}