2024 H2 Mathematics Paper 2 Question 1

Complex Numbers

Answers

$a = - 1,$ $b = - 4.$

$2 + 6 i .$

(a)

\begin{matrix} z = a z^{*} + b \\ - 2 + i = a (- 2 - i) + b \\ - 2 + i = - 2 a + b - a i \end{matrix}

Comparing imaginary parts,

\begin{aligned} 1 & = - a \\ a & = - 1 ∎ \end{aligned}

Comparing real parts,

\begin{aligned} - 2 & = - 2 a + b \\ = - 2 (- 1) + b \\ b & = - 4 ∎ \end{aligned}

(b)

\begin{aligned} w z - \frac{w}{z} \\ = (1 - 3 i) (- 2 + i) - \frac{1 - 3 i}{- 2 + i} \\ = (1 - 3 i) (- 2 + i) - \frac{(1 - 3 i) (- 2 - i)}{(- 2 + i) (- 2 - i)} \\ = - 2 + 7 i - 3 i^{2} - \frac{- 2 + 5 i + 3 i^{2}}{4 - i^{2}} \\ = 1 + 7 i - (- 1 + i) \\ = 2 + 6 i ∎ \end{aligned}