CAIE A-Level Mathematics A2 3.9.6 Complex Numbers Questions

Practise finding square roots of complex numbers in Cartesian and polar forms and solving related equations.

Exam points

find square roots of complex numbers in Cartesian form
use modulus and argument to express complex square roots
solve quadratic equations involving complex numbers

[Maximum number: 7]

The polynomial $x^{3}+5 x^{2}+31 x+75$ is denoted by p(x).

Hence find the complex numbers z which are roots of $\mathrm{p}\left(z^{2}\right)=0$ .

State $z_{1}=\sqrt{3} \mathrm{i}$ and $z_{2}=-\sqrt{3} \mathrm{i}$
Expand $(x+\mathrm{i} y)^{2}=-1+2 \sqrt{6} \mathrm{i}$ and compare real and imaginary parts
Marking guidance:

Allow for use of $z^{2}=-1-2 \sqrt{6}$ .

Obtain $x^{2}-y^{2}=-1$ and $x y=\sqrt{6}$
Solve to obtain x and y
Obtain $z_{3}=\sqrt{2}+\sqrt{3} \mathrm{i}$ and $z_{4}=-\sqrt{2}-\sqrt{3} \mathrm{i}$
Use $z^{2}=-1-2 \sqrt{6}$ i to obtain $z_{5}$ and $z_{6}$
Allow for use of $z^{2}=-1+2 \sqrt{6}$ .

Obtain $z_{5}=\sqrt{2}-\sqrt{3} \mathrm{i}$ and $z_{6}=-\sqrt{2}+\sqrt{3} \mathrm{i}$

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CAIE A-Level Mathematics A2 3.9.6 Complex Numbers Questions question 1