| Exam Board | AQA |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Direct nth roots: roots with geometric or algebraic follow-up |
| Difficulty | Standard +0.3 Part (a) is routine conversion to polar form requiring standard modulus/argument calculations. Part (b) requires understanding that vertices of an equilateral triangle centered at origin are related by rotations of 2π/3, which is a standard Further Maths technique. The question tests conceptual understanding of complex number geometry but follows a well-established pattern with no novel problem-solving required. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| 6(a) | Obtains correct modulus | |
| (allow √50) or argument. | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Allow √50 | 1.1b | B1 |
| Subtotal | 2 | |
| Q | Marking Instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 6(b) | Deduces moduli are all equal. | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| argument from part (a). | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Allow √50 | 1.1b | A1 |
| Subtotal | 3 | |
| Question total | 5 | |
| Q | Marking Instructions | AO |
Question 6:
--- 6(a) ---
6(a) | Obtains correct modulus
(allow √50) or argument. | 1.1b | B1 | 3π
i−
5 2e 4
Obtains completely correct
answer.
Allow √50 | 1.1b | B1
Subtotal | 2
Q | Marking Instructions | AO | Marks | Typical solution
--- 6(b) ---
6(b) | Deduces moduli are all equal. | 2.2a | M1 | 3π 2π π
− + =−
4 3 12
3π 4π 7π
− + =
4 3 12
π 7π
i− i
12 12
5 2e ,5 2e
2π
Adds a multiple of to their
3
argument from part (a). | 1.1a | M1
Obtains completely correct
solution.
Allow √50 | 1.1b | A1
Subtotal | 3
Question total | 5
Q | Marking Instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item Express $-5 - 5\text{i}$ in the form $re^{i\theta}$, where $-\pi < \theta \leq \pi$
[2 marks]
\item The point on an Argand diagram that represents $-5 - 5\text{i}$ is one of the vertices of an equilateral triangle whose centre is at the origin.
Find the complex numbers represented by the other two vertices of the triangle.
Give your answers in the form $re^{i\theta}$, where $-\pi < \theta \leq \pi$
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 2 2023 Q6 [5]}}