SPM.2.AHL.TZ0.8

pestleMathematicsAAHLPaper 2SPM· ahl-1-12-complex-numbers-cartesian-form-and-argand-diagsource ↗

The complex numbers $w$ and $z$ satisfy the equations

$\frac{w}{z} = 2{\text{i}}$

${{\text{z}}^ * } - 3w = 5 + 5{\text{i}}$ .

Find $w$ and $z$ in the form $a + b{\text{i}}$ where $a$ , ${\text{b}} \in \mathbb{Z}$ .

Markscheme / solution

substituting $w = 2{\text{i}}z$ into ${{\text{z}}^ * } - 3w = 5 + 5{\text{i}}$ M1

${{\text{z}}^ * } - 6{\text{i}}z = 5 + 5{\text{i}}$ A1

let $z = x + y{\text{i}}$

comparing real and imaginary parts of $\left( {x - y{\text{i}}} \right) - 6{\text{i}}\left( {x + y{\text{i}}} \right) = 5 + 5{\text{i}}$ M1

to obtain $x + 6y = 5$ and $- 6x - y = 5$ A1

attempting to solve for $x$ and $y$ ) M1

$x = - 1$ and $y = 1$ so $z = - 1 + {\text{i}}$ A1

hence $w = - 2 - 2{\text{i}}$ A1

[7 marks]

Examiners’ report

[N/A]