a) Either: Multiply numerator and denominator by $\left( {1 - 2i} \right)$, or equivalent

Obtain $ - 3i$

State modulus is 3

Refer to $u$ being on negative imaginary axis or equivalent and confirm argument as $ - \frac{1}{2}\pi $

Or: Using correct processes, divide moduli of numerator and denominator

Obtain 3

Subtract argument of denominator from argument of numerator

Obtain $ - {\tan ^{ - 1}}\frac{1}{2} - {\tan ^{ - 1}}2$ or $ - 0.464 - 1.107$ and hence $ - \frac{1}{2}\pi $ or $ - 1.57$

b) Show correct half-line from $u$ at angle $\frac{1}{4}\pi $ to real direction

Use correct trigonometry to find required value

Obtain $\frac{3}{2}\sqrt 2 $ or equivalent

c) Show, or imply, locus is a circle with centre $\left( {1 + i} \right)u$ and radius 1

Use correct method to find distance from origin to furthest point of circle

Obtain $3\sqrt 2  + 1$ or equivalent