Paper 1, Section I, 1B

Vectors and Matrices | Part IA, 2014

(a) Let

z=2+2iz=2+2 i

(i) Compute z4z^{4}.

(ii) Find all complex numbers ww such that w4=zw^{4}=z.

(b) Find all the solutions of the equation

e2z21=0e^{2 z^{2}}-1=0

(c) Let z=x+iy,zˉ=xiy,x,yRz=x+i y, \bar{z}=x-i y, x, y \in \mathbb{R}. Show that the equation of any line, and of any circle, may be written respectively as

Bz+Bˉzˉ+C=0 and zzˉ+Bˉz+Bzˉ+C=0B z+\bar{B} \bar{z}+C=0 \quad \text { and } \quad z \bar{z}+\bar{B} z+B \bar{z}+C=0

for some complex BB and real CC.

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