AS October 2020 Paper 1 Q3
3 In this question you must show detailed reasoning.
The complex number \(7 - 4\mathrm{i}\) is denoted by \(z\).
Find the following.
- \(|\omega|\)
- \(\arg(\omega)\), giving your answer correct to 3 significant figures [3]
| Scheme | Marks | AO |
|---|---|---|
| (i) DR \(3(7 - 4\mathrm{i}) - 4(7 + 4\mathrm{i}) = 21 - 12\mathrm{i} - 28 - 16\mathrm{i}\) | M1 | 1.1 |
| \(= -7 - 28\mathrm{i}\) | A1 | 1.1 |
| [2] | ||
| (ii) DR \((7 - 4\mathrm{i} + 1 - 3\mathrm{i})^2 = (8 - 7\mathrm{i})^2 = 64 - 112\mathrm{i} + 49\mathrm{i}^2\) | M1 | 1.1 |
| \(15 - 112\mathrm{i}\) | A1 | 1.1 |
| [2] | ||
| (iii) DR \(\dfrac{z + 1}{z - 1} = \dfrac{7 - 4\mathrm{i} + 1}{7 - 4\mathrm{i} - 1} = \dfrac{8 - 4\mathrm{i}}{6 - 4\mathrm{i}} \times \dfrac{6 + 4\mathrm{i}}{6 + 4\mathrm{i}}\) | M1 | 1.1 |
| \(= \dfrac{48 + 32\mathrm{i} - 24\mathrm{i} + 16}{36 + 16} = \dfrac{64}{52} + \dfrac{8}{52}\mathrm{i}\) | A1 | 1.1 |
| [2] |
Notes
(a)(i)
M1: \(z^*\) correct and brackets opened. Allow sign mistake with \(-16\mathrm{i}\), but not \(z/z^*\) mix-up
(a)(ii)
M1: Simplifying bracket and three term expansion. Allow \(z\ z^*\) mix-up here
(a)(iii)
M1: Simplifying and correct process for “realising” the denominator
Allow M1 if \(\dfrac{z^* + 1}{z^* - 1}\) used correctly
A1: Allow \(\frac{64 + 8\mathrm{i}}{52} = \frac{16 + 2\mathrm{i}}{13}\)
| Scheme | Marks | AO |
|---|---|---|
| DR \(\sqrt{7^2 + (-4)^2} = \sqrt{65}\) | B1 | 1.1 |
| \(\tan^{-1}\left(\pm\dfrac{4}{7}\right)\) | M1 | 1.1 |
| \(\sqrt{65}\,(\cos(-0.519) + \mathrm{i}\sin(-0.519))\) | A1 | 2.5 |
| [3] |
Notes
M1: Allow \(\tan^{-1}\left(\dfrac{7}{4}\right)\) if it is clear that this being used correctly (eg from diagram) to find the argument
A1: or \(\sqrt{65}\operatorname{cis}(-0.519)\) or \([\sqrt{65}, -0.519]\) or \(\sqrt{65}\operatorname{cis}(5.76)\) etc
Must be in the correct form ie c + is not c – is. If using \([r, \theta]\) then square brackets must be seen.
| Scheme | Marks | AO |
|---|---|---|
| DR 3 | B1ft | 1.1 |
| \(0.5 - -0.519\) | M1 | 1.1 |
| awrt 1.02 | A1ft | 1.1 |
| [3] |
Notes
B1ft: \(\sqrt{585} \div\) their \(|z|\)
M1: Use of \(\arg(z_1z_2) = \arg z_1 + \arg z_2\)
Must be seen
A1ft: \(0.5 -\) their \(\arg(z)\) in \([0, \pi/2]\)