20N.1.AHL.TZ0.H_4

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

substituting $z=x+\text{i}y$ and $z*=x-\text{i}y$        M1

$\frac{2\left(x+\text{i}y\right)}{3-\left(x-\text{i}y\right)}=\text{i}$

$2x+2\text{i}y=-y+\text{i}\left(3-x\right)$      

equate real and imaginary:        M1

$y=-2x$  AND  $2y=3-x$       A1

Note: If they multiply top and bottom by the conjugate, the equations $6x-2x^2+2y^2=0$ and $6y-4xy={\left(3-x\right)}^2+y^2$ may be seen. Allow for A1.

solving simultaneously:

$x=-1, y=2 \left(z=-1+2\text{i}\right)$      A1A1

[5 marks]

[N/A]