| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2023 |
| Session | January |
| Marks | 10 |
| Topic | Complex numbers 2 |
| Type | Find conjugate roots from polynomial |
| Difficulty | Standard +0.8 This is a further maths complex numbers question requiring knowledge that complex roots come in conjugate pairs for polynomials with real coefficients, then using this to find all three roots and determine unknown coefficients. While systematic, it requires multiple steps: identifying the conjugate root, finding the third root via sum of roots or factorization, plotting on an Argand diagram, and calculating p and q. The conceptual demand is moderate but above standard A-level, typical of FM content. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation |
$$f(z) = 3z^3 + pz^2 + 57z + q$$
where $p$ and $q$ are real constants.
Given that $3 - 2\sqrt{2}i$ is a root of the equation $f(z) = 0$
\begin{enumerate}[label=(\alph*)]
\item show all the roots of $f(z) = 0$ on a single Argand diagram, [7]
\item find the value of $p$ and the value of $q$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2023 Q8 [10]}}