If z=1 (1-i)(2+3i) , then |z|= - Vidyadip

MCQ

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

A
$1$
✓
$\frac{1}{\sqrt{26}}$
C
$\frac{5}{\sqrt{26}}$
D
none of these.

✓

Answer

Correct option: B.

$\frac{1}{\sqrt{26}}$

Let $\text{z}=\frac{1}{(1-\text{i})(2+3\text{i})}$
$\Rightarrow\text{z}=\frac{1}{2+\text{i}-3\text{i}^2}$
$\Rightarrow\text{z}=\frac{1}{2+\text{i}+3}$
$\Rightarrow\text{z}=\frac{1}{5+\text{i}}\times\frac{5-\text{i}}{5-\text{i}}$
$\Rightarrow\text{z}=\frac{5-\text{i}}{25-\text{i}^2}$
$\Rightarrow\text{z}=\frac{5-\text{i}}{25+1}$
$\Rightarrow\text{z}=\frac{5-\text{i}}{26}$
$\Rightarrow\text{z}=\frac{5}{26}-\frac{\text{i}}{26}$
$\Rightarrow|\text{z}|=\sqrt{\frac{25}{676}+\frac{1}{676}}$
$\Rightarrow\text{z}=\frac{1}{\sqrt{26}}$

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