Question

The modulus of the complex number \(\left( {\frac{{3 + 4i}}{{1 - 2i}}} \right)\) is 
1. 5
2. \(\sqrt 5\)
3. \(\frac{1}{{\sqrt 5 }}\)
4. 1/5

Answer 1

Correct Answer - Option 2 : \(\sqrt 5\)

Concept:

For a complex number z = x + iy, 

The modulus is given by

\(|z| = \sqrt {{x^2} + {y^2}} \)

the argument is given by 

\(θ = \arg z = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)

Calculation:

Given:

\(\left( {\frac{{3 + 4i}}{{1 - 2i}}} \right)\)

\(\left( {\frac{{3 \;+ \;4i}}{{1 \;- \;2i}}} \right)\times\left(\frac{1\;+ \;2i}{1\; +\; 2i} \right)\)

\(\left( {\frac{{3 \;+ \;6i\;+\;4i\;-8}}{{5}}} \right)\)

\(\left( {\frac{{-5 \;+ \;10i}}{{5}}} \right)=-1\;+\;2i\)

The modulus is given by

\(|z| = \sqrt {{x^2} + {y^2}} \)

\(|z| = \sqrt {{{(-1)}^2} + {2^2}} =\sqrt{5}\)