A2 October 2021 Paper 2 Q1
1. Given that
\[\begin{aligned} z_1 &= 3\left(\cos\left(\frac{\pi}{3}\right) + \mathrm{i}\sin\left(\frac{\pi}{3}\right)\right) \\ z_2 &= \sqrt{2}\left(\cos\left(\frac{\pi}{12}\right) - \mathrm{i}\sin\left(\frac{\pi}{12}\right)\right) \end{aligned}\](a) write down the exact value of
(i) \(|z_1 z_2|\)
(ii) \(\arg(z_1 z_2)\) (2)
(i) \(|z_1 z_2|\)
(ii) \(\arg(z_1 z_2)\) (2)
Given that \(w = z_1 z_2\) and that \(\arg(w^n) = 0\), where \(n \in \mathbb{Z}^+\)
(b) determine
(i) the smallest positive value of \(n\)
(ii) the corresponding value of \(|w^n|\) (3)
(i) the smallest positive value of \(n\)
(ii) the corresponding value of \(|w^n|\) (3)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(|z_1 z_2| = 3\sqrt{2}\) | B1 | 1.1b |
| (ii) \(\arg(z_1 z_2) = \dfrac{\pi}{3} + \left(-\dfrac{\pi}{12}\right) = \dfrac{\pi}{4}\) o.e. | B1 | 1.1b |
| (2) |
Notes
(a)(i)
B1: Deduces \(|z_1 z_2| = 3\sqrt{2}\)
(ii)
B1: Deduces \(\arg(z_1 z_2) = \dfrac{\pi}{4}\) o.e.
These marks may be awarded for \(z_1 z_2 = 3\sqrt{2}\left(\cos\dfrac{\pi}{4} + \mathrm{i}\sin\dfrac{\pi}{4}\right)\)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(n = 8\) | B1ft | 2.2a |
| (ii) \(|w^n| = \left(\text{their } |z_1 z_2|\right)^{\text{their } n}\) | M1 | 1.1b |
| \(|w^n| = 104\,976\) | A1 | 1.1b |
| (3) | ||
| (5 marks) |
Notes
(b)(i)
B1ft: \(2\pi\) divided by their \(\arg(z_1 z_2)\) found in part (a)(ii) to give an integer. Alternatively smallest positive integer multiple required to make their argument a multiple of \(2\pi\)
(ii)
M1: Their answer to (a)(i) to the power of their \(n\)
A1: 104 976