Question

The modulus of $\frac{1-i}{3+i}+\frac{4i}{5}$ is

• A. $\sqrt{5}$ unit
• B. $\frac{\sqrt{11}}{5}$ unit
• C. $\frac{\sqrt{5}}{5}$ unit
• D. $\frac{\sqrt{12}}{5}$ unit

Answer 1

Answer: (C)

$\frac{\sqrt{5}}{5}$ unit

Let $z=\frac{1-i}{3+i}+\frac{4i}{5}$
$=\frac{5-5i+12i-4}{5(3+i)}=\frac{1+7i}{5(3+i)}$
$=\frac{(1+7i)(3-i)}{5(9+1)}$
$=\frac{10+20i}{50}=\frac{1+2i}{5}$
$\therefore \left|z\right|=\sqrt{{\left(\frac{1}{5}\right)}^2+{\left(\frac{2}{5}\right)}^2}=\frac{1}{5}\sqrt{1+4}=\frac{\sqrt{5}}{2}$