Question 1416

Complex Numbers

2016 Paper 1 Question 7 Variant

Question

Do not use a calculator in answering this question.

(a)

Verify that

{- 1 + 4 \mathrm{i}}

is a root of the equation

w^2 + \left( - 2 - 3 \mathrm{i} \right) w + \left( 1 + 13 \mathrm{i} \right) = 0.

Hence, or otherwise, find the second root of the equation in cartesian form,

{p+\mathrm{i}q,}

showing your working.

[5]

(b)

The equation

z^3 - 3 z^2 + 7 z+k=0,

where

{k}

is a real constant, has a root

{z=1 + a \mathrm{i},}

where

{a}

is a positive real constant.

Find the values of

{a}

and

{k,}

showing your working.

[5]