A2 October 2021 Paper 1 Q3
3 In this question you must show detailed reasoning.
The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = -2 + 2\mathrm{i}\) and \(z_2 = 2\left(\cos\frac{1}{6}\pi + \mathrm{i}\sin\frac{1}{6}\pi\right)\).
(a) Find the modulus and argument of \(z_1\). [2]
(b) Hence express \(\dfrac{z_1}{z_2}\) in exact modulus-argument form. [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(|z_1| = \sqrt{8}\) | B1 | 1.1 |
| \(\arg(z_1) = \dfrac{3\pi}{4}\) | E1 | 1.1 |
| [2] |
Notes
E1: Must see some reasoning for \(\arg(z)\)
| Scheme | Marks | AO |
|---|---|---|
| DR \(\left|\dfrac{z_1}{z_2}\right| = \dfrac{\sqrt{8}}{2} = \sqrt{2}\) | B1 | 1.1 |
| \(\arg\left(\dfrac{z_1}{z_2}\right) = \dfrac{3\pi}{4} - \dfrac{\pi}{6} = \dfrac{7\pi}{12}\) | M1 A1 | 1.1 1.1 |
| so \(\dfrac{z_1}{z_2} = \sqrt{2}\left(\cos\dfrac{7\pi}{12} + \mathrm{i}\sin\dfrac{7\pi}{12}\right)\) | B1 | 1.1 |
| [4] |
Notes
M1: \(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)\) used