A2 October 2020 Paper 2 Q8
8 In this question you must show detailed reasoning.
The complex number \(-4 + \mathrm{i}\sqrt{48}\) is denoted by \(z\).
The points which represent the cube roots of \(z\) are denoted by \(A\), \(B\) and \(C\) and these form a triangle in an Argand diagram.
| Scheme | Marks | AO |
|---|---|---|
| DR \(r^2 = (-4)^2 + \left(\sqrt{48}\right)^2\) or (\(r\cos\theta = -4\) and \(r\sin\theta = \sqrt{48}\)) or \(\tan\theta = -\sqrt{3}\) oe | M1 | 2.1 |
| \(r = 8\) (ie \(z = 8\mathrm{e}^{\mathrm{i}\theta}\)) \(\theta = 2\pi/3\) (ie \(z = r\mathrm{e}^{\mathrm{i}2\pi/3}\)) | A1 | 1.1 |
| \(\sqrt[3]{8}\) or 2 | B1ft | 2.1 |
| \(\dfrac{2\pi}{9}\) soi | B1ft | 2.1 |
| \(\dfrac{2\pi}{3} + 2\pi k\) for \(k = 1\) and 2 oe seen | M1 | 2.2a |
| \(2\mathrm{e}^{\frac{2}{9}\pi\mathrm{i}},\ 2\mathrm{e}^{\frac{8}{9}\pi\mathrm{i}}\) and \(2\mathrm{e}^{-\frac{4}{9}\pi\mathrm{i}}\) | A1 | 1.1 |
| [6] |
Notes
M1: Correct use of relevant formula(e). Some working must be seen.
Correct answer with no working: M0A0
A1: Not \(\pm 8\) unless later corrected
or eg \(\theta = 8\pi/3\)
B1ft: Modulus of cube root(s) is the cube root of their modulus
B1ft: Argument of (principal) cube root is one third of their argument
M1: Considering further arguments at angular distance \(2\pi\)
A1: or eg \(2\mathrm{e}^{\frac{2}{9}\pi\mathrm{i}},\ 2\mathrm{e}^{\frac{8}{9}\pi\mathrm{i}}\) and \(2\mathrm{e}^{\frac{14}{9}\pi\mathrm{i}}\)
Must be in exponential form, not just \(r =\) and \(\theta =\). Do not condone any missing i’s.
| Scheme | Marks | AO |
|---|---|---|
| DR The cube roots form an equilateral triangle which has (3) lines of symmetry, (one) through each vertex | B1 | 2.2a |
| \(\theta = \dfrac{2\pi}{9},\ \theta = \dfrac{8\pi}{9}\) and \(\theta = -\dfrac{4\pi}{9}\) soi | B1 B1 | 2.2a 2.2a |
| [3] |
Notes
B1 B1: for one; for all three without extras
ft their angles if \(2\pi/3\) apart.
If valid alternatives, must come from clear explanation/diagram