A2 June 2023 Paper 2 Q6

(a) Express \(-5 - 5\mathrm{i}\) in the form \(r\mathrm{e}^{\mathrm{i}\theta}\), where \(-\pi \lt \theta \leqslant \pi\) [2 marks]

(b) The point on an Argand diagram that represents \(-5 - 5\mathrm{i}\) is one of the vertices of an equilateral triangle whose centre is at the origin.

Find the complex numbers represented by the other two vertices of the triangle.

Give your answers in the form \(r\mathrm{e}^{\mathrm{i}\theta}\), where \(-\pi \lt \theta \leqslant \pi\) [3 marks]

Mark scheme (a)

Scheme	Marks	AO
Obtains correct modulus (allow \(\sqrt{50}\)) or argument.	B1	1.1b
Obtains completely correct answer. Allow \(\sqrt{50}\)	B1	1.1b
	(2)

Typical solution

\[5\sqrt{2}\,\mathrm{e}^{\mathrm{i}\left(-\frac{3\pi}{4}\right)}\]

Mark scheme (b)

Scheme	Marks	AO
Deduces moduli are all equal.	M1	2.2a
Adds a multiple of \(\dfrac{2\pi}{3}\) to their argument from part (a).	M1	1.1a
Obtains completely correct solution. Allow \(\sqrt{50}\)	A1	1.1b
	(3)
	(5 marks)

Typical solution

\[-\frac{3\pi}{4} + \frac{2\pi}{3} = -\frac{\pi}{12}\]\[-\frac{3\pi}{4} + \frac{4\pi}{3} = \frac{7\pi}{12}\]\[5\sqrt{2}\,\mathrm{e}^{\mathrm{i}\left(-\frac{\pi}{12}\right)},\ 5\sqrt{2}\,\mathrm{e}^{\mathrm{i}\left(\frac{7\pi}{12}\right)}\]