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)

Let $z=\frac{1-i}{3+i}+\frac{4i}{5} $

$=\frac{5-5i+12i-4}{5\left(3+i\right)}=\frac{1+7i}{5\left(3+i\right)}$

$=\frac{\left(1+7i\right)\left(3-i\right)}{5\left(9+1\right)}$

$=\frac{10+20i}{50}=\frac{1+2i}{5}$

$\therefore \left|z\right|=\sqrt{\left(\frac{1}{5}\right)^{2} +\left(\frac{2}{5}\right)^{2}}$

$=\frac{1}{5} \sqrt{1+4}=\frac{\sqrt{5}}{5}$