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.