Question

The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\log }_{e}2\right)\ge {\log }_{\sqrt{2}}|5\sqrt{7}+9i|,$ $i=\sqrt{-1},$ is equal to :

• A. $2$
• B. $3$
• C. $8$
• D. $\sqrt{5}$

Answer 1

The correct option is B $3$

$\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\log }_{e}2\right)\ge {\log }_{\sqrt{2}}|5\sqrt{7}+9i|$
$\Rightarrow 2^{\frac{(|z|+3)(|z|-1)}{|z|+1}}\ge 2^3$
$\Rightarrow \frac{(|z|+3)(|z|-1)}{|z|+1}\ge 3$
$\Rightarrow |z{|}^2+2|z|-3\ge 3|z|+3$
$\Rightarrow |z{|}^2-|z|-6\ge 0$
$\Rightarrow (|z|-3)(|z|+2)\ge 0$
$\therefore |z{|}_{min}=3$