Question 1418

Complex Numbers

2018 Paper 2 Question 2 Variant

Question

(a)

One of the roots of the equation

9 x^4 + 60 x^3 + s x^2 + 114 x + t=0,

where

{s}

and

{t}

are real, is

{- 3 - 3 \mathrm{i}.}

Find the other roots of the equation and the values of

{s}

and

{t.}

[5]

(b)

The complex number

{w}

is such that

{w^3 = 125.}

(bi)

Given that one possible value of

{w}

{5,}

use a non-calculator method to find the other possible values of

{w.}

Give your answers in the form

{a+\mathrm{i}b,}

where

{a}

and

{b}

are exact values.

[3]

(bii)

Write these values of

{w}

in modulus-argument form and represent them on an Argand diagram.

[2]

(biii)

Find the sum and product of all the possible values of

{w,}

simplifying your answers.

[2]