A2 June 2022 Paper 1 Q11

OCR MEICurrent spec8 marksComplex Numbers De Moivre's Theorem

11 An Argand diagram with the point A representing a complex number \(z_1\) is shown below.

Argand diagram with axes Re and Im crossing at O; the point A representing z1 is in the first quadrant

The complex numbers \(z_2\) and \(z_3\) are \(z_1\mathrm{e}^{\frac{2}{3}\mathrm{i}\pi}\) and \(z_1\mathrm{e}^{\frac{4}{3}\mathrm{i}\pi}\) respectively.

(a)

(i) On the copy of the Argand diagram below, mark the points B and C representing the complex numbers \(z_2\) and \(z_3\). [2]

Copy of the Argand diagram from the Printed Answer Booklet: axes Re and Im crossing at O, with the point A representing z1 in the first quadrant

(ii) Show that \(z_1 + z_2 + z_3 = 0\). [2]

(b) Given now that \(z_1\), \(z_2\) and \(z_3\) are roots of the equation \(z^3 = 8\mathrm{i}\), find these three roots, giving your answers in the form \(a + \mathrm{i}b\), where \(a\) and \(b\) are real and exact. [4]

Argand diagram: A (z1), B (z2) and C (z3) on a circle centre O, forming an approximately equilateral triangle drawn with dashed lines; B in the second quadrant and C just below the origin, slightly left of the imaginary axis — Mark scheme (a)

Scheme	Marks	AO
(i)	M1 A1	1.1 1.1
	[2]

Scheme	Marks	AO
(ii) \(z_1 + z_2 + z_3 = z_1\left(1 + e^{2i\pi/3} + e^{4i\pi/3}\right)\) \(= z_1\dfrac{1 - e^{2i\pi}}{1 - e^{2i\pi/3}} = 0\)	M1 A1	2.1 2.2a
	[2]

Scheme	Marks
\(z_1 + z_2 + z_3 =\) \(z_1\left[1 + \cos\dfrac{2\pi}{3} + \cos\dfrac{4\pi}{3} + i\left(\sin\dfrac{2\pi}{3} + \sin\dfrac{4\pi}{3}\right)\right]\)	M1
\(= z_1\left[1 - \dfrac{1}{2} - \dfrac{1}{2} + i\left(\dfrac{\sqrt{3}}{2} - \dfrac{\sqrt{3}}{2}\right)\right] = 0\)	A1
	[2]

Mark scheme (b)

Scheme	Marks	AO
\(z^3 = 8\left(\cos\dfrac{1}{2}\pi + i\sin\dfrac{1}{2}\pi\right)\) \(= 8e^{i\pi/2}\)	B1	3.1a
\(\|z\| = 2\)	B1	1.1
\(z = 2e^{i\pi/6},\ 2e^{5i\pi/6},\ 2e^{3i\pi/2}\) \(= \sqrt{3} + i,\ -\sqrt{3} + i,\ -2i\)	B1B1	1.1
	[4]

Notes

B1: Exponential or modulus argument form soi

B1: Soi

B1B1: B1B0 if two out of three roots given

Notes

Notes

Alternative solution

Notes