| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| 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 finding the third root and using it to determine p and q. While it involves multiple steps (identifying conjugate root, finding third root via sum of roots or factorization, calculating p and q), these are standard techniques for FM students. The Argand diagram component is routine. Slightly above average difficulty due to the algebraic manipulation required, but follows a well-established method. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02k 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 Pure 2025 Q6 [10]}}