Question

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

• A. 3
• B. $\sqrt{5}$
• C. 2
• D. 8

Answer 1

Answer: (A)

exp $\left(\frac{(|z|+3)(|z|-1)}{|| z|+1|} \ln 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i|$
$\Rightarrow 2^{\frac{(|z|+3)(|z|-1)}{(|z|+1)} \geq \log _{\sqrt{2}}(16)}$
$\Rightarrow 2^{\frac{(|z|+3)(|z|-1)}{(|z|+1)} \geq 2^{3}}$
$\Rightarrow \frac{(|z|+3)(|z|-1)}{(|z|+1)} \geq 3$
$\Rightarrow (|z|+3)(|z|-1) \geq 3(|z|+1)$
$|z|^{2}+2|z|-3 \geq 3|z|+3$
$\Rightarrow |z|^{2}+|z|-6 \geq 0$
$\Rightarrow (|z|-3)(|z|+2) \geq 0$
$\Rightarrow |z|-3 \geq 0$
$\Rightarrow |z| \geq 3 \Rightarrow |z|_{\min }=3$