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

1. $\frac{1}{13}$

Solution:

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

$\Rightarrow\text{z}=\frac{1}{4+9\text{i}^2+12\text{i}}$

$\Rightarrow\text{z}=\frac{1}{4-9+12\text{i}}$

$\Rightarrow\text{z}=\frac{1}{-5+12\text{i}}$

$\Rightarrow\text{z}=\frac{1}{-5+12\text{i}}\times\frac{-5-12\text{i}}{-5-12\text{i}}$

$\Rightarrow\text{z}=\frac{-5-12\text{i}}{25+144}$

$\Rightarrow\text{z}=\frac{-5}{169}-\frac{12\text{i}}{169}$

$\Rightarrow|\text{z}|=\sqrt{\frac{25}{169^2}+\frac{144}{169^2}}$

$\Rightarrow|\text{z}|=\frac{1}{\sqrt{169}}$

$\Rightarrow|\text{z}|=\frac{1}{13}$