| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Geometric properties in Argand diagram |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question testing standard modular form operations and Argand diagram geometry. Parts (a) and (b) are direct applications of multiplication/division rules for modulus and argument. Part (c) is routine plotting. Part (d) requires recognizing that P, Q, R lie on a circle of radius 2 centered at the origin, then using the inscribed angle theoremβa standard technique for this topic. While it requires multiple steps and geometric insight, it follows predictable patterns for Further Maths complex numbers questions. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| 8(a) | States | 1.1b |
| Answer | Marks |
|---|---|
| 8(b) | Obtains4 |
| Answer | Marks | Guidance |
|---|---|---|
| β12 | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 8(c) | β1.83(2595715β¦) |
| Answer | Marks | Guidance |
|---|---|---|
| π§π§1 | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| π§π§2 | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 8(d) | Recognises that P, Q and R are points on the circumference of a circle |
| Answer | Marks | Guidance |
|---|---|---|
| Or recognises is a circle. | 3.1a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Correctly dedu | cπ€π€es | =the2 angle as |
| Answer | Marks | Guidance |
|---|---|---|
| ππ ππ 2 4 | 2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| πππ π οΏ½ππ πππποΏ½ππ | 2.4 | E1 |
| Total | 7 | 7ππ |
| Answer | Marks | Guidance |
|---|---|---|
| Q | Marking instructions | AO |
Question 8:
--- 8(a) ---
8(a) | States | 1.1b | B1
--- 8(b) ---
8(b) | Obtains4
7ππ
Accept
β12 | 1.1b | B1 | 4
ππ 3ππ 7ππ
--- 8(c) ---
8(c) | β1.83(2595715β¦)
Draws a point (or line from O) labelled P (or ) in the first quadrant.
π§π§1 | 1.1b | B1 | β = β
6 4 12
Draws a point (or line from O) labelled Q (or ) in the second quadrant.
π§π§2 | 1.1b | B1
--- 8(d) ---
8(d) | Recognises that P, Q and R are points on the circumference of a circle
β possibly implied by circle drawn on diagram.
Or recognises is a circle. | 3.1a | B1 | 3ππ ππ 7ππ
πππποΏ½ππ = 4 β6 = 12
7ππ
ππ(aπ
π
οΏ½nππgle= at ciΓ·rc2umference angle at centre)
12
1
= 2Γ
Correctly dedu|cπ€π€es| =the2 angle as
7ππ
Accept any exact equivalent an π
π
οΏ½g le, e. g . or 52.5Β°
ππ ππ 2 4 | 2.2a | B1
0.2916Μππ
Explains why the angle is half that of the angle for any position of R.
Award 3/3 for a complete and correct algebraic solution.
πππ
π
οΏ½ππ πππποΏ½ππ | 2.4 | E1
Total | 7 | 7ππ
=
24
Q | Marking instructions | AO | Marks | Typical solutionGiven that $z_1 = 2\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)$ and $z_2 = 2\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $|z_1z_2|$ [1 mark]
\item Find the value of $\arg\left(\frac{z_1}{z_2}\right)$ [1 mark]
\item Sketch $z_1$ and $z_2$ on the Argand diagram below, labelling the points as $P$ and $Q$ respectively. [2 marks]
\item A third complex number $w$ satisfies both $|w| = 2$ and $-\pi < \arg w < 0$
Given that $w$ is represented on the Argand diagram as the point $R$, find the angle $PRQ$.
Fully justify your answer. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2019 Q8 [7]}}