| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 9 |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Challenging +1.3 Part (a) requires converting a complex number to polar form and finding cube roots using de Moivre's theorem—a standard Further Maths technique but with moderately awkward arithmetic (√2 - √6i). Part (b) combines complex numbers with linear transformations and area scaling, requiring knowledge that area scales by |det(M)|. While this tests multiple topics, the individual steps are routine for Further Maths students, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 4.02r nth roots: of complex numbers4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $z^3 = \sqrt{2} - \sqrt{6}i$, giving your answers in the form $re^{i\theta}$ where $r > 0$ and $0 \leq \theta < 2\pi$ [5 marks]
\item The transformation represented by the matrix $\mathbf{M} = \begin{bmatrix} 5 & 1 \\ 1 & 3 \end{bmatrix}$ acts on the points on an Argand Diagram which represent the roots of the equation in part (a).
Find the exact area of the shape formed by joining the transformed points. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q8 [9]}}