AS June 2019 Paper 1 Q1
1 You are given that \(z = 3 - 4\mathrm{i}\).
- \(|z|\),
- \(\arg(z)\),
- \(z^*\). [3]
On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w^*\) is represented by the point \(B\).
| Scheme | Marks | AO |
|---|---|---|
| \(|z| = 5\) | B1 | 1.1 |
| \(\arg z = -0.927\) rads or \(-53.1^\circ\) | B1 | 1.1 |
| \(z^* = 3 + 4\mathrm{i}\) | B1 | 1.1 |
| [3] |
Notes
B1: (1st) From \(\sqrt{3^2 + 4^2}\) or BC
B1: (2nd) Or 5.36 rads (5.35589…) or \(307^\circ\) (or 306.8698…)
From \(\tan^{-1}\left(-\tfrac{4}{3}\right)\), \(\tan\theta = \tfrac{4}{3}\) or BC
| Scheme | Marks | AO |
|---|---|---|
| \(A\) and \(B\) are reflections of each other... | M1 | 1.1 |
| ... in the real (or horizontal or \(x\)) axis | A1 | 1.1 |
| [2] |
Notes
M1: Reflection / Reflected
Allow references to \(w\) and \(w^*\) (or \(z\) and \(z^*\) etc) rather than \(A\) and \(B\). Do not allow “mirrored” unless also a reference to “reflection”
A1: Correct mirror line
\(y = 0\) ok
Do not allow “positive real axis”
Could describe the geometrical relationship in terms not involving the word “reflection” but would need to be entirely correct and un-ambiguous.
Diagram only is no marks. Diagram with accompanying description for a general case is fine.