A2 June 2023 Paper 1 Q5

OCR MEICurrent spec7 marksComplex Numbers De Moivre's Theorem

(a) In this question you must show detailed reasoning.
Determine the sixth roots of \(-64\), expressed in \(r\mathrm{e}^{\mathrm{i}\theta}\) form. [4]

(b) Represent the roots on the Argand diagram below.

Argand diagram from the Printed Answer Booklet: blank axes Re and Im crossing at the origin O

[3]

Mark scheme (a)

Scheme	Marks	AO
DR Let \(z = r(\cos\theta + \mathrm{i}\sin\theta)\) or \(z = r\mathrm{e}^{\mathrm{i}\theta}\) \(z^6 = 64(\cos\pi + \mathrm{i}\sin\pi)\) or \(64\mathrm{e}^{\mathrm{i}\pi}\)	M1	1.1
\(\Rightarrow r = 2\)	B1	1.1
\(z = 2\mathrm{e}^{\mathrm{i}\pi/6}, 2\mathrm{e}^{\mathrm{i}\pi/2}, 2\mathrm{e}^{5\mathrm{i}\pi/6}, 2\mathrm{e}^{-\mathrm{i}\pi/6}, 2\mathrm{e}^{-\mathrm{i}\pi/2}, 2\mathrm{e}^{-5\mathrm{i}\pi/6}\)	A1 A1	2.5 1.1
	[4]

Notes

M1: condone \(-\pi\) for \(\pi\)

B1: soi

A1: 1 correct

A1: all correct or \(2\mathrm{e}^{7\mathrm{i}\pi/6}, 2\mathrm{e}^{3\mathrm{i}\pi/2}, 2\mathrm{e}^{11\mathrm{i}\pi/6}\)

Argand diagram: six points equally spaced on a dotted circle of radius 2 centred at the origin, two of them on the imaginary axis at 2i and -2i — Mark scheme (b)