Determine the value of

(i)     $1 + \omega  + {\omega ^2}$;

(ii)     $1 + \omega {\text{*}} + {(\omega {\text{*}})^2}$.

Show that $(\omega  - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13$.

Find the values of $x$ that satisfy the equation $\left| p \right| = \left| q \right|$.

Solve the inequality $\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}$.

(i)     METHOD 1

$1 + \omega  + {\omega ^2} = \frac{{1 - {\omega ^3}}}{{1 - \omega }} = 0$    A1

as $\omega  \ne 1$     R1

METHOD 2

solutions of $1 - {\omega ^3} = 0$ are $\omega  = 1,{\text{ }}\omega {\text{ = }}\frac{{ - 1 \pm \sqrt 3 {\text{i}}}}{2}$     A1

verification that the sum of these roots is 0     R1

(ii)     $1 + \omega {\text{*}} + {(\omega {\text{*}})^2} = 0$     A2

[4 marks]

$(\omega  - 3{\omega ^2})({\omega ^2} - 3\omega ) =  - 3{\omega ^4} + 10{\omega ^3} - 3{\omega ^2}$    M1A1

EITHER

$=  - 3{\omega ^2}({\omega ^2} + \omega  + 1) + 13{\omega ^3}$    M1

$=  - 3{\omega ^2} \times 0 + 13 \times 1$    A1

OR

$=  - 3\omega  + 10 - 3{\omega ^2} =  - 3({\omega ^2} + \omega  + 1) + 13$    M1

$=  - 3 \times 0 + 13$    A1

OR

substitution by $\omega  = \frac{{ - 1 \pm \sqrt 3 {\text{i}}}}{2}$ in any form     M1

numerical values of each term seen     A1

THEN

$= 13$    AG

[4 marks]

$\left| p \right| = \left| q \right| \Rightarrow \sqrt {{1^2} + {3^2}}  = \sqrt {{x^2} + {{(2x + 1)}^2}}$    (M1)(A1)

$5{x^2} + 4x - 9 = 0$    A1

$(5x + 9)(x - 1) = 0$    (M1)

$x = 1,{\text{ }}x =  - \frac{9}{5}$    A1

[5 marks]

$pq = (1 - 3{\text{i}})\left( {x + (2x + 1){\text{i}}} \right) = (7x + 3) + (1 - x){\text{i}}$    M1A1

$\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2} \Rightarrow (7x + 3) + 8 < {(1 - x)^2}$    M1

$\Rightarrow {x^2} - 9x - 10 > 0$    A1

$\Rightarrow (x + 1)(x - 10) > 0$    M1

$x <  - 1,{\text{ }}x > 10$    A1

[6 marks]