| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Solve polynomial equations with complex roots |
| Difficulty | Challenging +1.2 This is a structured Further Maths question that guides students through standard complex number techniques (exponential form, conjugate relationships) to solve a quartic. While it requires multiple steps and involves FP3 content, the question provides significant scaffolding through parts (a)(i)-(iii), making the solution path clear. The algebraic manipulation is routine for Further Maths students, and the final step of solving a quadratic in cos θ is straightforward once the setup is complete. |
| Spec | 4.02d Exponential form: re^(i*theta)4.02j Cubic/quartic equations: conjugate pairs and factor theorem4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta) |
I appreciate your request, but I notice that the content you've provided for Question 6 only contains:
```
Question 6:
6
6
```
This doesn't appear to be actual mark scheme content — there are no marking annotations (M1, A1, B1, etc), marking criteria, or guidance notes to clean up.
Could you please provide the complete mark scheme content for Question 6? Once you share the full text with the actual criteria and marking points, I'll be happy to format it according to your specifications.6 It is given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that
$$z + \frac { 1 } { z } = 2 \cos \theta$$
\item Find a similar expression for
$$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$
(2 marks)
\item Hence show that
$$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$
(3 marks)
\end{enumerate}\item Hence solve the quartic equation
$$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$
giving the roots in the form $a + \mathrm { i } b$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 Q6 [17]}}