Question 1408a

Complex Numbers

2008 Paper 1 Question 8 Variant

Question

A graphic calculator is not to be used in answering this question.

(a)

It is given that

{z_1 = 1 + \sqrt{3} \mathrm{i}.}

Find the value of

{z_1^3,}

showing clearly how you obtain your answer.

[3]

(b)

Given that

{1 + \sqrt{3} \mathrm{i}}

is a root of the equation

2z^3 + az^2 + bz -20 = 0,

find the values of the real numbers

{a}

and

{b.}

[4]

(c)

For these values of

{a}

and

{b,}

solve the equation in part (ii), and show all the roots on an Argand diagram.

[4]