CAIE A-Level Mathematics 3.9.8 Complex-Number Loci

Practise converting complex equations and inequalities into lines, circles, half-planes and angle regions on Argand diagrams and finding extreme modulus or argument values.

Exam points

translate Re z, Im z, |z − a| and arg(z − a) into the corresponding geometric boundary
use strict or inclusive inequalities to select the correct side, disc or angular region
locate tangent or boundary points to calculate maximum or minimum |z| or arg z

[Maximum number: 4]

The complex number $-1+\sqrt{7} \mathrm{i}$ is denoted by u. It is given that u is a root of the equation

2 x^{3}+3 x^{2}+14 x+k=0,

where k is a real constant.

(a)

On an Argand diagram, sketch the locus of points representing complex numbers z satisfying the equation |z-u|=2.

[ 2 ]

Official markscheme Argand diagram for Q10(c).

|z-u|=2 is the circle of radius 2 centred at u. Since
$u=-1+\sqrt7\,i,$
the locus is a circle centred at $(-1,\sqrt7)$ on the Argand diagram with radius 2.

B1 for centre $-1+\sqrt7 i$ . B1 for a circle of radius 2, with scale or radius indicated; if more than one circle is shown, maximum B1.

(b)

Determine the greatest value of $\arg z$ for points on this locus, giving your answer in radians.

[ 2 ]

Carry out a complete method for calculating the maximum value of $\arg z$ for
correct circle

M1

e.g. $\frac{\pi}{2}+\tan ^{-1} \frac{1}{\sqrt{7}}+\frac{\pi}{4}$ Can be implied by $155.7^{\circ}$ .

Obtain answer 2.72 radians

A1

CAO. The question requires radians.

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3.9.8—Complex numbers question 1

Question (a)

Question (b)