| Exam Board | SPS |
|---|---|
| Module | SPS ASFM (SPS ASFM) |
| Year | 2020 |
| Session | May |
| Marks | 14 |
| Topic | Complex Numbers Arithmetic |
| Type | Given one complex root of cubic or quartic, find all roots |
| Difficulty | Standard +0.3 This is a structured complex numbers question with clear guidance (given root 2+i). Part (a) uses conjugate root theorem to find quadratic factors—routine for A-level Further Maths. Part (b) solves the quadratics using standard methods. Parts (c-d) involve moduli and sum of roots (Vieta's formulas), all straightforward applications. The 14 marks reflect length rather than conceptual difficulty; each step follows standard Further Maths techniques without requiring novel insight. |
| Spec | 4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k Argand diagrams: geometric interpretation |
\textbf{In this question you must show detailed reasoning.}
You are given that $f(z) = 4z^4 - 12z^3 + 41z^2 - 128z + 185$ and that $2 + \mathrm{i}$ is a root of the equation $f(z) = 0$.
\begin{enumerate}[label=(\alph*)]
\item Express $f(z)$ as the product of two quadratic factors with integer coefficients. [5]
\item Solve $f(z) = 0$. [3]
Two loci on an Argand diagram are defined by $C_1 = \{z:|z| = r_1\}$ and $C_2 = \{z:|z| = r_2\}$ where $r_1 > r_2$. You are given that two of the points representing the roots of $f(z) = 0$ are on $C_1$ and two are on $C_2$. $R$ is the region on the Argand diagram between $C_1$ and $C_2$.
\item Find the exact area of $R$. [4]
\item $\omega$ is the sum of all the roots of $f(z) = 0$.
Determine whether or not the point on the Argand diagram which represents $\omega$ lies in $R$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS ASFM 2020 Q3 [14]}}