Question 1414a

Complex Numbers

2014 Paper 1 Question 5 Variant

Question

It is given that

{z=1 + 2 \mathrm{i}.}

(a)

Without using a calculator, find the values of

{z^3}

and

\frac{1}{z^{2}}

in cartesian form

{x + \mathrm{i}y,}

showing your working.

[4]

(b)

The real numbers

{p}

and

{q}

are such that

pz^3 + \frac{q}{z^{2}}

is purely imaginary.

Find, in terms of

{p,}

the value of

{q}

and the value of

pz^3 + \frac{q}{z^{2}}.

[3]

Answer

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