Edexcel IAL Mathematics FP2.3 further complex numbers

Further complex questions link algebra with geometry: expand De Moivre forms, interpret Argand loci and transform circles or lines between planes.

Exam points

Apply De Moivre's theorem to derive cos nθ identities or solve polynomial equations.
Convert modulus or argument conditions into Argand lines, circles and regions.
Rearrange z-plane to w-plane mappings before finding centre, radius or equation.

Covered syllabus points

FP2.3.1 - Euler’s relation eiθ = cos θ + i sin θ FP2.3.2 - De Moivre’s theorem and its FP2.3.3 - Loci and regions in the Argand diagram FP2.3.4 - Transformations between the z-plane and w-plane

[Maximum number: 10]

(a)

Use de Moivre's theorem to show that

\sin 5 \theta \equiv 16 \sin ^{5} \theta-20 \sin ^{3} \theta+5 \sin \theta

[ 5 ]

$(\cos\theta+i\sin\theta)^5=\cos5\theta+i\sin5\theta$
B1
$\begin{aligned} &=\cos^5\theta+5\cos^4\theta(i\sin\theta)+10\cos^3\theta(i\sin\theta)^2\\ &\quad+10\cos^2\theta(i\sin\theta)^3+5\cos\theta(i\sin\theta)^4+(i\sin\theta)^5 \end{aligned}$
M1
$=\cos^5\theta+5i\sin\theta\cos^4\theta-10\sin^2\theta\cos^3\theta-10i\sin^3\theta\cos^2\theta+5\sin^4\theta\cos\theta+i\sin^5\theta$
A1
Equating imaginary parts,
$\sin5\theta=5\sin\theta\cos^4\theta-10\sin^3\theta\cos^2\theta+\sin^5\theta$ $=5\sin\theta(1-\sin^2\theta)^2-10\sin^3\theta(1-\sin^2\theta)+\sin^5\theta$
M1
$\sin5\theta=16\sin^5\theta-20\sin^3\theta+5\sin\theta$
A1*

(b)

Hence determine the five distinct solutions of the equation

16 x^{5}-20 x^{3}+5 x+\frac{1}{5}=0

giving your answers to 3 decimal places.

[ 5 ]

Let $x=\sin \theta \quad 16 x^{5}-20 x^{3}+5 x=-\frac{1}{5} \Rightarrow \sin 5 \theta=\ldots$

M1A1

$\Rightarrow \theta=\frac{1}{5} \sin ^{-1}\left( \pm \frac{1}{5}\right)=38.306$ (or -2.307,69.692.110.306,141.693,182.306 )
(or in radians $-0.0402 \ldots 0.6685 \ldots, 1.216 \ldots, 1.925 \ldots, 2.473 \ldots$ )

dM1

Two of (awrt) $x=\sin \theta=-0.963,-0.555,-0.040,0.620,0.938$

A1

All of (awrt) $x=\sin \theta=-0.963,-0.555,-0.040,0.620,0.938$

A1 (5)

[Maximum number: 8]

A complex number z is represented by the point P in an Argand diagram.

Given that

|z-2 i|=|z-3|

(a)

sketch the locus of P. You do not need to find the coordinates of any intercepts.

The transformation T from the z-plane to the w-plane is given by

w=\frac{\mathrm{i} z}{z-2 \mathrm{i}} \quad z \neq 2 \mathrm{i}

Given that T maps |z-2 i|=|z-3| to a circle C in the w-plane,

[ 2 ]

(b)

find the equation of C, giving your answer in the form

|w-(p+q i)|=r

where p, q and r are real numbers to be determined.

[ 6 ]

$w=\frac{iz}{z-2i}\Rightarrow z=\frac{2wi}{w-i}$
Let w=u+iv. Substitution into the locus of z and simplification gives
$\left(u+\frac6{13}\right)^2+\left(v-\frac9{13}\right)^2=\frac4{13}.$
Therefore
$\left|w-\left(-\frac6{13}+\frac9{13}i\right)\right|=\frac{2}{\sqrt{13}}.$
M1
dM1
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dddM1
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Covered syllabus points

FP2.3 - Further complex numbers question 1

Question (a)

Question (b)

FP2.3 - Further complex numbers question 2

Question (a)

Question (b)