| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring knowledge of complex conjugate roots, polynomial division/factorization, coefficient comparison, and Argand diagram geometry. While the individual techniques are standard (conjugate root theorem, finding the real root, calculating triangle area), the combination of steps and the geometric application elevates it above typical A-level questions but remains within expected FM1 scope. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
2.
$$f ( z ) = 2 z ^ { 3 } + p z ^ { 2 } + q z - 41$$
where $p$ and $q$ are integers.\\
The complex number $5 - 4 \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\begin{enumerate}[label=(\alph*)]
\item Write down another complex root of this equation.
\item Solve the equation $\mathrm { f } ( \mathrm { z } ) = 0$ completely.
\item Determine the value of $p$ and the value of $q$.
When plotted on an Argand diagram, the points representing the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ form the vertices of a triangle.
\item Determine the area of this triangle.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2024 Q2 [9]}}