| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | nth roots via factorization |
| Difficulty | Standard +0.8 This FP2 question requires students to verify an algebraic identity involving complex exponentials (routine), then apply it cleverly by recognizing that -1 = -2cos(2π/3) to factorize z^8 - z^4 + 1, and finally extract 8 roots using de Moivre's theorem. The key insight of matching coefficients to find θ = 2π/3 elevates this above standard root-finding exercises, though the techniques themselves are well-practiced at FP2 level. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| - Correct multiplication of brackets | M1 | |
| - \(e^{i\theta} + e^{-i\theta} = 2\cos\theta\) | A1 | 2 marks; Clearly shown |
| Answer | Marks | Guidance |
|---|---|---|
| - \(2\cos\theta = 1\) | M1 | |
| - \(\theta = \frac{\pi}{3}\) | A1 | |
| - \(z^4 = e^{\frac{\pi i}{3}}\) or \(e^{-\frac{\pi i}{3}}\) | M1 | |
| - \(z = e^{\frac{\pi i}{12}}, e^{\frac{7\pi i}{12}}, e^{\frac{5\pi i}{12}}, e^{\frac{11\pi i}{12}}\) | m1 | |
| - | A2, 1, 0F | 6 marks; Al if 3 roots correct |
| Answer | Marks | Guidance |
|---|---|---|
| - | B2, 1, 0 | B1 for 4 roots indicated correctly on a circle. CAO |
| - Indication that \(r = 1\) | B1 | 3 marks |
**(a)**
- Correct multiplication of brackets | M1 |
- $e^{i\theta} + e^{-i\theta} = 2\cos\theta$ | A1 | 2 marks; Clearly shown
- **Total: 2 marks**
**(b)**
- $2\cos\theta = 1$ | M1 |
- $\theta = \frac{\pi}{3}$ | A1 |
- $z^4 = e^{\frac{\pi i}{3}}$ or $e^{-\frac{\pi i}{3}}$ | M1 |
- $z = e^{\frac{\pi i}{12}}, e^{\frac{7\pi i}{12}}, e^{\frac{5\pi i}{12}}, e^{\frac{11\pi i}{12}}$ | m1 |
- | A2, 1, 0F | 6 marks; Al if 3 roots correct
- **Total: 6 marks**
**(c)**
- | B2, 1, 0 | B1 for 4 roots indicated correctly on a circle. CAO
- Indication that $r = 1$ | B1 | 3 marks
- **Total: 3 marks**
**Question 8 Total: 11 marks**
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# **GRAND TOTAL: 75 marks**8
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\left( z ^ { 4 } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { 4 } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 8 } - 2 z ^ { 4 } \cos \theta + 1$$
(2 marks)
\item Hence solve the equation
$$z ^ { 8 } - z ^ { 4 } + 1 = 0$$
giving your answers in the form $\mathrm { e } ^ { \mathrm { i } \phi }$, where $- \pi < \phi \leqslant \pi$.
\item Indicate the roots on an Argand diagram.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2009 Q8 [11]}}