Question 1412

Complex Numbers

2012 Paper 1 Question 6 Variant

Question

Do not use a calculator in answering this question.

The complex number

{z}

is given by

{z=c + 5 \mathrm{i},}

where

{c}

is a non-zero real number.

(i)

Find

{z^3}

in the form

{x+\mathrm{i}y.}

[2]

(ii)

Given that

{z^3}

is purely imaginary, find the possible values of

{z.}

[2]

(iii)

For the value of

{z}

found in part (ii) for which

{c<0,}

find the smallest positive integer

{n}

such that

{\left| z^n \right| > 1000.}

State the modulus and argument of

{z^n}

when

{n}

takes this value.

[4]