Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).

Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).

\(\left| {{z_1}} \right| = \sqrt {10} ;{\text{ }}\arg ({z_2}) = - \frac{{3\pi }}{4}{\text{ }}\left( {{\text{accept }}\frac{{5\pi }}{4}} \right)\)     A1A1

[2 marks]

\(\left| {{z_1} + \alpha{z_2}} \right| = \sqrt {{{(1 - \alpha )}^2} + {{(3 - \alpha )}^2}} \) or the squared modulus     (M1)(A1)

attempt to minimise \(2{\alpha ^2} - 8\alpha  + 10\) or their quadratic or its half or its square root     M1

obtain \(\alpha  = 2\) at minimum     (A1)

state \(\sqrt 2 \) as final answer     A1

[5 marks]

Disappointingly, few candidates obtained the correct argument for the second complex number, mechanically using arctan(1) but not thinking about the position of the number in the complex plane.

Most candidates obtained the correct quadratic or its square root, but few knew how to set about minimising it.