| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a standard Further Maths question on complex conjugate roots with real coefficients. Part (a) requires recalling that complex roots come in conjugate pairs (routine knowledge), part (b) is straightforward plotting, and part (c) involves expanding factors or using Vieta's formulas—all well-practiced techniques with no novel insight required. Slightly easier than average due to its predictable structure. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
| VILU SIHI NI JIIIM ION OC | VIUV SIHI NI III M M I ON OO | VIAV SIHI NI JIIIM I ION OC |
3.
$$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$
where $a , b , c$ and $d$ are integers.\\
The complex numbers $3 + \mathrm { i }$ and $- 1 - 2 \mathrm { i }$ are roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
\begin{enumerate}[label=(\alph*)]
\item Write down the other roots of this equation.
\item Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.
\item Determine the values of $a , b , c$ and $d$.
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VILU SIHI NI JIIIM ION OC & VIUV SIHI NI III M M I ON OO & VIAV SIHI NI JIIIM I ION OC \\
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\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2020 Q3 [9]}}