## Task A-2.2.

Determine the complex number $z$ such that

$$
\operatorname{Re} \frac{1}{1-z}=2 \quad \text { and } \quad \operatorname{Im} \frac{1}{1-z}=-1
$$
## First solution.

The given conditions can be written as one equation

$$
\frac{1}{1-z}=2-i
$$

from which it follows

$$
\begin{array}{rl}
1-z=\frac{1}{2-i} & 1 \text { point } \\
1-z=\frac{2}{5}+\frac{1}{5} i & 2 \text { points } \\
z=\frac{3}{5}-\frac{1}{5} i . & 1 \text { point }
\end{array}
$$