Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

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

Solution:

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

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

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

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

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

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

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

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

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

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