Question 1417

Complex Numbers

2017 Paper 1 Question 8 Variant

Question

Do not use a calculator in answering this question.

(a)

Find the roots of the equation

z^2 \left(1 - 2 \mathrm{i}\right) - 2 z + (2 + 4 \mathrm{i})=0,

giving your answers in cartesian form

{a+\mathrm{i}b.}

[3]

(bi)

Given that

{\omega=- 1 + \mathrm{i},}

find

{\omega^2,}

{\omega^3}

and

{\omega^4}

in cartesian form.

Given also that

\omega^4 + p \omega^3 + 23 \omega^2 + q \omega + 82=0,

where

{p}

and

{q}

are real, find

{p}

and

{q.}

[4]

(bii)

Using the values of

{p}

and

{q}

in part (bii), express

\omega^4 + p \omega^3 + 23 \omega^2 + q \omega + 82

as the product of two quadratic factors.

[3]