A2 June 2023 Paper 1 Q1
1
(a) The complex number \(a + \mathrm{i}b\) is denoted by \(z\).
(i) Write down \(z^*\). [1]
(ii) Find \(\mathrm{Re}(\mathrm{i}z)\). [2]
(b) The complex number \(w\) is given by \(w = \dfrac{5 + \mathrm{i}\sqrt{3}}{2 - \mathrm{i}\sqrt{3}}\).
(i) In this question you must show detailed reasoning.
Express \(w\) in the form \(x + \mathrm{i}y\). [2]
Express \(w\) in the form \(x + \mathrm{i}y\). [2]
(ii) Convert \(w\) to modulus-argument form. [2]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(a - \mathrm{i}b\) | B1 | 1.2 |
| [1] | ||
| (ii) \(\mathrm{i}z = -b + a\mathrm{i}\) | M1 | 1.1 |
| so \(\mathrm{Re}(\mathrm{i}z) = -b\) | A1 | 1.1 |
| [2] |
| Scheme | Marks | AO |
|---|---|---|
| (i) DR \(\dfrac{5 + \sqrt{3}\mathrm{i}}{2 - \sqrt{3}\mathrm{i}} = \dfrac{(5 + \sqrt{3}\,\mathrm{i})(2 + \sqrt{3}\,\mathrm{i})}{(2 - \sqrt{3}\,\mathrm{i})(2 + \sqrt{3}\,\mathrm{i})}\) | M1 | 1.1a |
| \(= \dfrac{7 + 7\sqrt{3}\mathrm{i}}{7}\) \(= 1 + \sqrt{3}\mathrm{i}\) | A1 | 1.1 |
| [2] | ||
| (ii) \(\left|1 + \sqrt{3}\mathrm{i}\right| = 2\) \(\arg(1 + \sqrt{3}\,\mathrm{i}) = \dfrac{\pi}{3}\) | B1ft | 1.1 |
| so \(w = 2\left(\cos\frac{\pi}{3} + \mathrm{i}\sin\frac{\pi}{3}\right)\) | B1 | 1.1 |
| [2] |
Notes
(b)(i)
A1: an intermediate step must be seen before final answer
(b)(ii)
B1ft: for either their modulus or argument soi
B1: cao. Allow \(60^\circ\)