| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Factor theorem and finding roots |
| Difficulty | Standard +0.3 This is a straightforward Further Pure F1 question requiring standard techniques: substituting to find p, polynomial division/factorisation to find complex roots, plotting on an Argand diagram, and calculating distances. While it involves complex numbers and multiple parts, each step follows routine procedures with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.
$$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$
Given that $z = 3$ is a root of the equation $f ( z ) = 0$\\
(a) show that $p = - 87$\\
(b) Use algebra to determine the other roots of $\mathrm { f } ( \mathrm { z } ) = 0$, giving your answers in simplest form.
On an Argand diagram
\begin{itemize}
\item the root $z = 3$ is represented by the point $P$
\item the other roots of $\mathrm { f } ( \mathrm { z } ) = 0$ are represented by the points $Q$ and $R$
\item the number $z = - 9$ is represented by the point $S$\\
(c) Show on a single Argand diagram the positions of $P , Q , R$ and $S$\\
(d) Determine the perimeter of the quadrilateral $P Q S R$, giving your answer as a simplified surd.
\end{itemize}
\hfill \mbox{\textit{Edexcel F1 2024 Q2 [9]}}