Question:
If $z=\frac{1}{(1-i)(2+3 i)}$, than $|z|=$
(a) 1
(b) $1 / \sqrt{26}$
(c) $5 / \sqrt{26}$
(d) none of these
Solution:
(b) $1 / \sqrt{26}$
Let $z=\frac{1}{(1-i)(2+3 i)}$
$\Rightarrow z=\frac{1}{2+i-3 i^{2}}$
$\Rightarrow z=\frac{1}{2+i+3}$
$\Rightarrow z=\frac{1}{5+i} \times \frac{5-i}{5-i}$
$\Rightarrow z=\frac{5-i}{25-i^{2}}$
$\Rightarrow z=\frac{5-i}{25+1}$
$\Rightarrow z=\frac{5-i}{26}$
$\Rightarrow z=\frac{5}{26}-\frac{i}{26}$
$\Rightarrow|z|=\sqrt{\frac{25}{676}+\frac{1}{676}}$
$\Rightarrow z=\frac{1}{\sqrt{26}}$