| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Moderate -0.3 This is a structured Further Maths question with clear steps guiding students through standard complex number techniques (simplification, modulus, exponential form, Argand diagram). Part (a) requires multiplying by conjugate (routine), parts (b-c) are direct applications of formulas, and part (e) uses geometric insight but is heavily scaffolded by the diagram in part (d). While it's Further Maths content, the question is more procedural than problem-solving, making it slightly easier than average A-level difficulty. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1+i}{1-i} = \frac{(1+i)^2}{1-i^2} = i\) | M1A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \( | z_2 | = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1 = |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = 1\) | B1 | PI |
| \(\theta = \frac{1}{2}\pi,\ \frac{1}{3}\pi\) | B1B1 | Deduct 1 mark if extra solutions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Diagram showing \(z_1\), \(z_2\), \(z_1+z_2\) | B2,1F | Positions of the 3 points relative to each other must be approximately correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\arg(z_1 + z_2) = \frac{5}{12}\pi\) | B1 | Clearly shown |
| \(\tan\frac{5}{12}\pi = \frac{1 + \frac{1}{2}\sqrt{3}}{\frac{1}{2}}\) | M1 | Allow if B1 earned |
| \(= 2 + \sqrt{3}\) | A1 | AG, must earn B1 for this |
# Question 3:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1+i}{1-i} = \frac{(1+i)^2}{1-i^2} = i$ | M1A1 | AG |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|z_2| = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1 = |z_1|$ | M1A1 | |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = 1$ | B1 | PI |
| $\theta = \frac{1}{2}\pi,\ \frac{1}{3}\pi$ | B1B1 | Deduct 1 mark if extra solutions |
## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Diagram showing $z_1$, $z_2$, $z_1+z_2$ | B2,1F | Positions of the 3 points relative to each other must be approximately correct |
## Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\arg(z_1 + z_2) = \frac{5}{12}\pi$ | B1 | Clearly shown |
| $\tan\frac{5}{12}\pi = \frac{1 + \frac{1}{2}\sqrt{3}}{\frac{1}{2}}$ | M1 | Allow if B1 earned |
| $= 2 + \sqrt{3}$ | A1 | AG, must earn B1 for this |
---3 The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by
$$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$
\begin{enumerate}[label=(\alph*)]
\item Show that $z _ { 1 } = \mathrm { i }$.
\item Show that $\left| z _ { 1 } \right| = \left| z _ { 2 } \right|$.
\item Express both $z _ { 1 }$ and $z _ { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
\item Draw an Argand diagram to show the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 1 } + z _ { 2 }$.
\item Use your Argand diagram to show that
$$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2006 Q3 [12]}}