Question

For the complex number $z=\frac{3+\sqrt{-1}}{2-\sqrt{-1}}$, the correct option(s) is/are

• A. $\text{Re}\left(z\right)=1$
• B. $\text{Im}\left(z\right)=1$
• C. $\text{Im}\left(z\right)=0$
• D. $|z|=\sqrt{2}$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\text{Re}\left(z\right)=1$
B $\text{Im}\left(z\right)=1$
D $|z|=\sqrt{2}$

$z=\frac{3+\sqrt{-1}}{2-\sqrt{-1}}=\frac{3+i}{2-i}\Rightarrow \frac{3+i}{2-i}\times \frac{2+i}{2+i}$
$\Rightarrow \frac{(3+i)(2+i)}{(2-i)(2+i)}$
$=\frac{6+i^2+5i}{4-i^2}=\frac{5+5i}{5}=1+i$

$\therefore \text{Re}\left(z\right)=1$ and $\text{Im}\left(z\right)=1$
Also, $|z|=\sqrt{1^2+1^2}=\sqrt{2}$