Question

The set of points in $C$ satisfying the inequality $\left|\arg \right(z)-\frac{\pi }{2}|<\frac{\pi }{2}$ is

• A. $\{z:\text{Im}\left(z\right)>0\}$
• B. $\{z:\text{Im}\left(z\right)<0\}$
• C. $\{z:\text{Re}\left(z\right)>0\}$
• D. $\{z:\text{Re}\left(z\right)<0\}$

Answer 1

The correct option is A $\{z:\text{Im}\left(z\right)>0\}$

Given that $\left|\text{arg}\right(z)-\frac{\pi }{2}|<\frac{\pi }{2}$
$\Rightarrow -\frac{\pi }{2}<\text{arg}\left(z\right)-\frac{\pi }{2}<\frac{\pi }{2}$
$\Rightarrow 0<\text{arg}\left(z\right)<\pi$
$\Rightarrow z$ lies in either first quadrant or second quadrant and hence $\text{Im}\left(z\right)>0$
$\therefore$ the required set is $\{z:\text{Im}\left(z\right)>0\}$