The modulus of 1- i 3+ i +4 i 5 is - Vidyadip

MCQ

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

A
$\sqrt{5}$ units
B
$\frac{\sqrt{11}}{5}$ units
✓
$\frac{\sqrt{5}}{5}$ units
D
$\frac{\sqrt{12}}{5}$ units

✓

Answer

Correct option: C.

$\frac{\sqrt{5}}{5}$ units

(C)
$\left|\frac{1- i }{3+ i }+\frac{4 i }{5}\right|=\left|\frac{(1- i )(3- i )}{(3+ i )(3- i )}+\frac{4 i }{5}\right|$
$=\left|\frac{3-1-4 i}{9-(-1)}+\frac{4 i}{5}\right|$
$=\left|\frac{2(1-2 i)}{10}+\frac{4 i}{5}\right|$
$=\left|\frac{1}{5}+\frac{2}{5} i \right|$
$=\sqrt{\left(\frac{1}{5}\right)^2+\left(\frac{2}{5}\right)^2}=\sqrt{\frac{5}{25}}=\frac{\sqrt{5}}{5}$ units

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