Question:
The least value of $|z|$ where $z$ is complex number which satisfies the inequality
$\exp \left(\frac{(|z|+3)(|z|-1)}{|| z|+1|} \log _{e} 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i|$
$\mathrm{i}=\sqrt{-1}$, is equal to :
Correct Option: 1
Solution:
$\exp \left(\frac{(|z|+3)(|z|-1)}{|| z|+1|} \ln 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i|$
$\Rightarrow 2^{\frac{(|z|+3)(|z|-1)}{(|z|+1)}} \geq \log _{\sqrt{2}}(16)$
$\Rightarrow 2^{\frac{(|z|+3)(\mid z-1)}{(|z|+1)}} \geq 2^{3}$
$\Rightarrow \frac{(|z|+3)(|z|-1)}{(|z|+1)} \geq 3$
$\Rightarrow(|z|+3)(|z|-1) \geq 3(|z|+1)$
$|z|^{2}+2|z|-3 \geq 3|z|+3$
$\Rightarrow|z|^{2}+|z|-6 \geq 0$
$\Rightarrow(|z|-3)(|z|+2) \geq 0 \Rightarrow|z|-3 \geq 0$
$\Rightarrow|z| \geq 3 \quad \Rightarrow|z|_{\min }=3$