Question

If $z=\sqrt{\frac{1+i}{1-i}},$ then the modulus of $z$ is

• A. $1$
• B. $2$
• C. $\sqrt{2}$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $1$

$z=\sqrt{\frac{1+i}{1-i}}\Rightarrow z=\frac{\sqrt{1+i}}{\sqrt{1-i}}\times \frac{\sqrt{1+i}}{\sqrt{1+i}}\Rightarrow z=\frac{1+i}{\sqrt{1-i^2}}\Rightarrow z=\frac{1+i}{\sqrt{2}}$
$\Rightarrow z=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$

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