Consider the complex numbers $z = 1 + 2{\text{i}}$ and $w = 2 + a{\text{i}}$ , where $a \in \mathbb{R}$ .

Find a when

(a)     $\left| w \right| = 2\left| z \right|;$ ;

(b)     ${\text{Re}}(zw) = 2\operatorname{Im} (zw)$ .

(a)     $\left| z \right| = \sqrt 5$ and $\left| w \right| = \sqrt {4 + {a^2}}$

$\left| w \right| = 2\left| z \right|$

$\sqrt {4 + {a^2}} = 2\sqrt 5$

attempt to solve equation     M1

Note: Award M0 if modulus is not used.

$a = \pm 4$     A1A1     N0

(b)     $zw = (2 - 2a) + (4 + a){\text{i}}$     A1

forming equation $2 - 2a = 2(4 + a)$     M1

$a = - \frac{3}{2}$     A1     N0

[6 marks]

Most candidates made good attempts to answer this question. Weaker candidates did not get full marks due to difficulties recognizing the notation and working with modulus of a complex number.