Question

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

• A. $\frac{1}{13}$
• B. $\frac{1}{5}$
• C. $\frac{1}{12}$
• D. none of these

Answer 1

The correct option is A

$\frac{1}{13}$

$\text{Let}z=\frac{1}{(2+3i{)}^2}\Rightarrow z=\frac{1}{4+9i^2+12i}\Rightarrow z=\frac{1}{4-9+12i}\Rightarrow z=\frac{1}{-5+12i}\Rightarrow z=\frac{1}{-5+12i}\times \frac{-5-12i}{-5-12i}\Rightarrow z=\frac{-5-12i}{25+144}\Rightarrow z=\frac{-5}{169}-\frac{12i}{169}\Rightarrow |z|=\sqrt{\frac{25}{{169}^2}+\frac{144}{{169}^2}}\Rightarrow |z|=\frac{1}{\sqrt{169}}\Rightarrow |z|=\frac{1}{13}$