AS June 2019 Paper 1 Q8
8 Given that \(z_1 = 2\left(\cos\dfrac{\pi}{6} + \mathrm{i}\sin\dfrac{\pi}{6}\right)\) and \(z_2 = 2\left(\cos\dfrac{3\pi}{4} + \mathrm{i}\sin\dfrac{3\pi}{4}\right)\)
(a) Find the value of \(|z_1z_2|\) [1 mark]
(b) Find the value of \(\arg\left(\dfrac{z_1}{z_2}\right)\) [1 mark]
(c) Sketch \(z_1\) and \(z_2\) on the Argand diagram below, labelling the points as \(P\) and \(Q\) respectively. [2 marks]
(d) A third complex number \(w\) satisfies both \(|w| = 2\) and \(-\pi \lt \arg w \lt 0\)
Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(P\hat{R}Q\).
Fully justify your answer. [3 marks]
Given that \(w\) is represented on the Argand diagram as the point \(R\), find the angle \(P\hat{R}Q\).
Fully justify your answer. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| States 4 | B1 | 1.1b |
Typical solution
\[4\]| Scheme | Marks | AO |
|---|---|---|
| Obtains \(-\frac{7\pi}{12}\) Accept \(-1.83(2595715\ldots)\) | B1 | 1.1b |
Typical solution
\[\frac{\pi}{6} - \frac{3\pi}{4} = -\frac{7\pi}{12}\]| Scheme | Marks | AO |
|---|---|---|
| Draws a point (or line from \(O\)) labelled \(P\) (or \(z_1\)) in the first quadrant. | B1 | 1.1b |
| Draws a point (or line from \(O\)) labelled \(Q\) (or \(z_2\)) in the second quadrant. | B1 | 1.1b |
Typical solution
| Scheme | Marks | AO |
|---|---|---|
| Recognises that \(P\), \(Q\) and \(R\) are points on the circumference of a circle – possibly implied by circle drawn on diagram. Or recognises \(|w| = 2\) is a circle. | B1 | 3.1a |
| Correctly deduces the angle \(P\hat{R}Q\) as \(\frac{7\pi}{24}\) Accept any exact equivalent angle, e.g. \(0.291\dot{6}\pi\) or \(52.5^\circ\) | B1 | 2.2a |
| Explains why the angle \(P\hat{R}Q\) is half that of the angle \(P\hat{O}Q\) for any position of \(R\). Award 3/3 for a complete and correct algebraic solution. | E1 | 2.4 |
| (7 marks) |
Typical solution
