AS October 2021 Paper 1 Q7
7
(a)
(i) Find the modulus and argument of \(z_1\), where \(z_1 = 1 + \mathrm{i}\). [2]
(ii) Given that \(|z_2| = 2\) and \(\arg(z_2) = \tfrac{1}{6}\pi\), express \(z_2\) in \(a + b\mathrm{i}\) form, where \(a\) and \(b\) are exact real numbers. [2]
(b) Using these results, find the exact value of \(\sin\tfrac{5}{12}\pi\), giving the answer in the form \(\dfrac{\sqrt{m} + \sqrt{n}}{p}\), where \(m\), \(n\) and \(p\) are integers. [5]
| Scheme | Marks | AO |
|---|---|---|
| (i) \(|z_1| = \sqrt{2}\) | B1 | 1.1 |
| \(\arg(z_1) = \tfrac{1}{4}\pi\) | B1 | 1.1 |
| [2] | ||
| (ii) \(z_2 = 2\left(\cos\tfrac{1}{6}\pi + \mathrm{i}\sin\tfrac{1}{6}\pi\right)\) | M1 | 1.1 |
| \(= \sqrt{3} + \mathrm{i}\) | A1 | 1.1 |
| [2] |
Notes
(a)(i)
B1: (1st) 1.41 or better
B1: (2nd) allow \(45^\circ\)
(a)(ii)
A1: 1.73 or better
| Scheme | Marks | AO |
|---|---|---|
| \(z_1z_2 = (1 + \mathrm{i})(\sqrt{3} + \mathrm{i})\) | M1 | 3.1a |
| \(= \sqrt{3} - 1 + (\sqrt{3} + 1)\mathrm{i}\) | A1 | 1.1 |
| \(z_1z_2 = 2\sqrt{2}\left(\cos\tfrac{5}{12}\pi + \mathrm{i}\sin\tfrac{5}{12}\pi\right)\) | M1 | 3.1a |
| So \(\sin\tfrac{5}{12}\pi = \dfrac{\sqrt{3} + 1}{2\sqrt{2}}\) | A1 | 1.1 |
| \(= \dfrac{\sqrt{2}(\sqrt{3} + 1)}{4} = \dfrac{\sqrt{6} + \sqrt{2}}{4}\) | A1 | 3.2a |
| [5] |
Notes
M1: (1st) Finding \(z_1z_2\) (using cartesian form)
M1: (2nd) Use of \(\arg(z_1z_2) = \arg z_1 + \arg z_2\)