Question

If $z=\frac{1}{(1-i)(2+3i)}$, then $|z|=$

• A. $1$
• B. $1/\sqrt{26}$
• C. $5/\sqrt{26}$
• D. $noneofthese$

Answer 1

The correct option is B $1/\sqrt{26}$

$z=\frac{1}{(1-i)(2+3i)}$

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

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

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

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

Therefore,

$\left|z\right|=\sqrt{{\left(\frac{5}{26}\right)}^2+{\left(\frac{-1}{26}\right)}^2}$

$\Rightarrow \left|z\right|=\sqrt{\frac{25+1}{{\left(26\right)}^2}}$

$\Rightarrow \left|z\right|=\frac{1}{\sqrt{26}}$