| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Moderate -0.3 This is a standard Further Maths question on complex conjugate roots with real coefficients. Part (a) requires immediate recall of the conjugate root theorem, part (b) involves routine expansion and coefficient comparison or substitution, and part (c) is straightforward plotting. While it's Further Maths content, it follows a completely standard template with no novel problem-solving required, making it slightly easier than average overall. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| VIIN SIHILNI III IM ION OC | VIAV SIHI NI III HM ION OO | VIAV SIHI NI III IM I ON OC |
4.
$$f ( z ) = 2 z ^ { 3 } - z ^ { 2 } + a z + b$$
where $a$ and $b$ are integers.
The complex number $- 1 - 3 \mathrm { i }$ 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 Determine the value of $a$ and the value of $b$.
\item Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.
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VIIN SIHILNI III IM ION OC & VIAV SIHI NI III HM ION OO & VIAV SIHI NI III IM I ON OC \\
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\hfill \mbox{\textit{Edexcel F1 2021 Q4 [7]}}