A2 June 2023 Paper 1 Q3
3
(a) Show that \(\dfrac{-3 + \sqrt{3}\,\mathrm{i}}{2} = \sqrt{3}\,\mathrm{e}^{\frac{5}{6}\pi\mathrm{i}}\). [2]
(b) Hence determine the exact roots of the equation \(z^5 = \dfrac{9\left(-3 + \sqrt{3}\,\mathrm{i}\right)}{2}\), giving the roots in the form \(r\mathrm{e}^{\mathrm{i}\theta}\) where \(r \gt 0\) and \(0 \leqslant \theta \lt 2\pi\). [3]
| Scheme | Marks |
|---|---|
| \(r = \dfrac{1}{2}\sqrt{(-3)^2 + \left(\sqrt{3}\right)^2} = \dfrac{1}{2}\sqrt{12} = \sqrt{3}\) | B1 |
| \(\arctan\left(-\dfrac{\sqrt{3}}{3}\right) = -\dfrac{\pi}{6}\) \(\Rightarrow \theta = -\dfrac{\pi}{6} + \pi = \dfrac{5\pi}{6}\) | B1 |
| [2] |
Notes
B1: AG so must show use of \(|z| = \sqrt{a^2 + b^2}\)
B1: AG. Or \(\theta = \pi - \arctan\left(\frac{\sqrt{3}}{3}\right)\), may be indicated on a diagram, but clear reasoning must be shown (eg. finding complementary angle, or use of Pythagoras’ theorem and then arcsin or arccos)
Alternative method
| Scheme | Marks |
|---|---|
| \(\sqrt{3}\mathrm{e}^{\frac{5}{6}\pi\mathrm{i}} = \sqrt{3}\left(\cos\left(\dfrac{5}{6}\pi\right) + \mathrm{i}\sin\left(\dfrac{5}{6}\pi\right)\right)\) | M1 |
| \(= \sqrt{3}\left(-\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}\mathrm{i}\right) = \dfrac{\sqrt{3}\mathrm{i} - 3}{2}\) | A1 |
A1: AG Clearly shown
| Scheme | Marks |
|---|---|
| \(r = \left(9\sqrt{3}\right)^{\frac{1}{5}} = \left(3^{\frac{5}{2}}\right)^{\frac{1}{5}} = \sqrt{3}\) | B1 |
| \(\theta = \dfrac{1}{5}\left(\dfrac{5}{6}\pi + 2r\pi\right) = \dfrac{\pi}{30}(5 + 12r)\) for \(r = 0, 1, 2, 3, 4\) | M1 |
| \(\Rightarrow z = \sqrt{3}\mathrm{e}^{\frac{1}{6}\pi\mathrm{i}}, \sqrt{3}\mathrm{e}^{\frac{17}{30}\pi\mathrm{i}}, \sqrt{3}\mathrm{e}^{\frac{29}{30}\pi\mathrm{i}}, \sqrt{3}\mathrm{e}^{\frac{41}{30}\pi\mathrm{i}}, \sqrt{3}\mathrm{e}^{\frac{53}{30}\pi\mathrm{i}}\) | A1 |
| [3] |
Notes
B1: For \(r = \sqrt{3}\) oe (including 1.73....)
M1: For their \(\frac{5}{6}\pi + 2\pi n\) from (a) divided by 5 (either in terms of \(n\), or for at least two values of \(n\)).
A1: Allow \(\sqrt{3}\mathrm{e}^{\frac{1}{30}(5 + 12n)\pi\mathrm{i}}\) for \(n = 0, 1, 2, 3, 4\).
Accept only \(r = \sqrt{3}\) or \((3)^{\frac{1}{2}}\)
For last two marks, If M0 then SC B1 for all five roots