AS June 2018 Paper 1 Q9
9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by
\[\left\{z : |z| \leqslant 4\sqrt{2}\right\} \cap \left\{z : \tfrac{1}{4}\pi \leqslant \arg z \leqslant \tfrac{1}{3}\pi\right\}.\]
(i) Find, in modulus-argument form, the complex number represented by the point P. [2]
(ii) Find, in the form \(a + \mathrm{i}b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
(iii) In this question you must show detailed reasoning.
Determine whether the points representing the complex numbers
Determine whether the points representing the complex numbers
- \(3 + 5\mathrm{i}\)
- \(5.5(\cos 0.8 + \mathrm{i}\sin 0.8)\)
| Scheme | Marks | AO |
|---|---|---|
| \(\arg z = \tfrac{1}{4}\pi,\ |z| = 4\sqrt{2}\) | B1 | 1.1 |
| so \(z = 4\sqrt{2}\left(\cos\tfrac{1}{4}\pi + \mathrm{i}\sin\tfrac{1}{4}\pi\right)\) | B1 | 2.5 |
| [2] |
Notes
B1: (2nd) allow \(45^\circ\) for both marks
| Scheme | Marks | AO |
|---|---|---|
| \(\arg z = \tfrac{1}{3}\pi,\ |z| = 4\sqrt{2}\) | M1 | 1.1a |
| so \(z = 4\sqrt{2}\left(\cos\tfrac{1}{3}\pi + \mathrm{i}\sin\tfrac{1}{3}\pi\right)\) | A1 | 1.1 |
| \(= 2\sqrt{2} + 2\sqrt{6}\mathrm{i}\) | A1 | 1.1 |
| [3] |
Notes
A1: (2nd) o.e., must be exact
| Scheme | Marks | AO |
|---|---|---|
| DR \(|3 + 5\mathrm{i}| = \sqrt{9 + 25} = \sqrt{34}\) | M1 | 3.1a |
| \(\sqrt{34} \gt \sqrt{32}\) so not in the region | A1 | 2.1 |
| \(5.5(\cos 0.8 + \mathrm{i}\sin 0.8)\) has mod 5.5 and arg 0.8 \(5.5 \lt \sqrt{32} = 5.656\ldots\) | M1 | 3.1a |
| \(0.785 \lt 0.8 \lt 1.047\) so is in the region | A1 | 2.1 |
| [4] |
Notes
M1: (1st) finding modulus (correct method)
M1: (2nd) comparing mod or argument
A1: (2nd) checking both conditions