| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Find conjugate roots from polynomial |
| Difficulty | Standard +0.3 This is a standard Further Maths question on complex conjugate roots with straightforward algebraic steps. Part (a) requires knowing that complex roots come in conjugate pairs (basic recall), part (b) involves forming a quadratic factor from the conjugate pair and polynomial division to find remaining roots (routine technique), and part (c) is plotting points. While it requires multiple steps and is Further Maths content, it follows a well-practiced algorithm without requiring problem-solving insight, making it slightly easier than average. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j 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. Solutions relying on calculator technology are not acceptable.
\end{enumerate}
Given that $x = 2 + 3 \mathrm { i }$ is a root of the equation
$$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$
(a) write down another complex root of this equation.\\
(b) Use algebra to determine the other 2 roots of the equation.\\
(c) Show all 4 roots on a single Argand diagram.
\hfill \mbox{\textit{Edexcel F1 2023 Q2 [7]}}