Question

Locate the complex number $z=x+iy$ for which $lo{g}_{1/3}\left\{lo{g}_{1/2}\right(|z{|}^2+4|z|+3\left)\right\}<0$

• A. Empty set
• B. circle of radius 6 and center at origin
• C. circle of radius 3 and center at origin
• D. circle of radius 2 and center at origin

Answer 1

The correct option is A Empty set

${\log }_{1/3}\left\{{\log }_{1/2}\right(|z{|}^2+4|z|+3\left)\right\}<0$
$\Rightarrow {\log }_{1/2}(|z{|}^2+4|z|+3)>0$
$\Rightarrow |z{|}^2+4|z|+3<0$
$\Rightarrow (\left|z\right|+1)(\left|z\right|+3)<0$
Since, $\left|z\right|+1>0$ and $\left|z\right|+3>0$
Therefore, $z$ belongs to empty set.
Ans: A