# Solve the following

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

$\Rightarrow z=\frac{1}{2+i-3 i^{2}}$

$\Rightarrow z=\frac{1}{2+i+3}$

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

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

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

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

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

$\Rightarrow|z|=\sqrt{\frac{25}{676}+\frac{1}{676}}$

$\Rightarrow z=\frac{1}{\sqrt{26}}$