| 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 real polynomials, then using factor theorem or polynomial division to find real constants. Part (a) requires plotting three roots on an Argand diagram (routine visualization), while part (b) involves algebraic manipulation with complex numbers. The multi-step nature and Further Maths content place it above average difficulty, but the techniques are standard for this level. |
| 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 $3 - 2\sqrt{21}i$ 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]}}