| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Factored form to roots |
| Difficulty | Moderate -0.8 This is a straightforward Further Pure 1 question requiring factorization of two quadratics (one difference of squares pattern, one completing the square) to find complex roots, then plotting them. While complex numbers are an FP1 topic, the actual techniques are routine: solving quadratics and basic Argand diagram plotting with no geometric insight or proof required. The 6 total marks and two-part structure confirm this is below-average difficulty even for Further Maths. |
| Spec | 4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((4x^2 + 9) = 0 \Rightarrow x = \frac{3i}{2}, \frac{-3i}{2}\) | M1 | An attempt to solve \((4x^2 + 9) = 0\) which involves \(i\). |
| A1 | \(\frac{3i}{2}, \frac{-3i}{2}\) | |
| \((x^2 - 2x + 5) = 0 \Rightarrow x = \frac{2 \pm \sqrt{4-4(1)(5)}}{2(1)} = \frac{2 \pm \sqrt{-16}}{2} \Rightarrow x = 1 \pm 2i\) | M1 | Solves the 3TQ |
| A1 | \(1 \pm 2i\) | [4] |
| (b) Any two of their roots plotted correctly on a single diagram, which have been found in part (a). Both sets of their roots plotted correctly on a single diagram with symmetry about \(y = 0\). | B1 ft |
**(a)** $(4x^2 + 9) = 0 \Rightarrow x = \frac{3i}{2}, \frac{-3i}{2}$ | M1 | An attempt to solve $(4x^2 + 9) = 0$ which involves $i$.
| A1 | $\frac{3i}{2}, \frac{-3i}{2}$
$(x^2 - 2x + 5) = 0 \Rightarrow x = \frac{2 \pm \sqrt{4-4(1)(5)}}{2(1)} = \frac{2 \pm \sqrt{-16}}{2} \Rightarrow x = 1 \pm 2i$ | M1 | Solves the 3TQ
| A1 | $1 \pm 2i$ | [4]
**(b)** Any two of their roots plotted correctly on a single diagram, which have been found in part (a). Both sets of their roots plotted correctly on a single diagram with symmetry about $y = 0$. | B1 ft | | [2]
**Total: [6]**
---$$f(x) = (4x^2 + 9)(x^2 - 2x + 5)$$
\begin{enumerate}[label=(\alph*)]
\item Find the four roots of $f(x) = 0$ [4]
\item Show the four roots of $f(x) = 0$ on a single Argand diagram. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2013 Q4 [6]}}