Question

$\mathrm{If}z=\frac{1}{(1-i)(2+3i)},\mathrm{than}\left|z\right|=$
(a) 1
(b) $1/\sqrt{26}$
(c) $5/\sqrt{26}$
(d) none of these

Answer 1

(b) $\frac{1}{\sqrt{26}}$
$$\begin{gathered} \mathrm{Let}z=\frac{1}{\left(1-i\right)\left(2+3i\right)} \\ \Rightarrow z=\frac{1}{2+i-3i^2} \\ \Rightarrow z=\frac{1}{2+i+3} \end{gathered}$$

$\Rightarrow z=\frac{1}{5+i}\times \frac{5-i}{5-i}$

$\Rightarrow z=\frac{5-i}{25-i^2}$

$\Rightarrow z=\frac{5-i}{25+1}$

$\Rightarrow z=\frac{5-i}{26}$

$\Rightarrow z=\frac{5}{26}-\frac{i}{26}$

$\Rightarrow \left|z\right|=\sqrt{{\displaystyle \frac{25}{676}+\frac{1}{676}}}$
$\Rightarrow z=\frac{1}{\sqrt{26}}$