AS October 2020 Paper 1 Q2

OCR MEICurrent spec4 marksComplex Numbers

2 Fig. 2 shows two complex numbers \(z_1\) and \(z_2\) represented on an Argand diagram.

Fig. 2: Argand diagram with axes Re and Im meeting at O; z1 is in the first quadrant, a little to the right of the Im axis and well above the Re axis; z2 is in the first quadrant, further to the right and a little above the Re axis — Fig. 2

(a) On a copy of Fig. 2, mark points representing each of the following complex numbers.

\(z_1^*\)
\(z_2 - z_1\) [2]

(b) In this question you must show detailed reasoning.
In the case where \(z_1 = 1 + 2\mathrm{i}\) and \(z_2 = 3 + \mathrm{i}\), find \(\dfrac{z_2 - z_1}{z_1^*}\) in the form \(a + \mathrm{i}b\), where \(a\) and \(b\) are real numbers. [2]

Scheme	Marks	AO
	B1 B1	1.1 1.1
	[2]

Mark scheme (b)

Scheme	Marks	AO
DR \(\dfrac{z_2 - z_1}{z_1^*} = \dfrac{2 - \mathrm{i}}{1 - 2\mathrm{i}} = \dfrac{(2 - \mathrm{i})(1 + 2\mathrm{i})}{(1 - 2\mathrm{i})(1 + 2\mathrm{i})}\)	M1	1.1
\(= \dfrac{4}{5} + \dfrac{3}{5}\mathrm{i}\)	A1	1.1
	[2]

Notes

M1: \(\times\) top and bottom by \(1 + 2\mathrm{i}\)

A1: condone \(\dfrac{4 + 3\mathrm{i}}{5}\)