Question

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

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

Answer 1

The correct option is B

$3$

Explanation of the correct option.

Compute the required value.

Given : $\exp \left(\frac{\left(\left|\mathrm{z}\right|+3\right)\left(\left|\mathrm{z}\right|-1\right)}{\left(\left|\mathrm{z}\right|+1\right)}{\log }_e\left(2\right)\right)\ge {\log }_{\sqrt{2}}\left(\left|5\sqrt{7}+9\mathrm{i}\right|\right)$

$\Rightarrow$ $2^{\frac{\left(\left|\mathrm{z}\right|+3\right)\left(\left|\mathrm{z}\right|-1\right)}{\left(\left|\mathrm{z}\right|+1\right)}}\ge {\log }_{\sqrt{2}}\left(16\right)$

$\Rightarrow$ $2^{\frac{\left(\left|\mathrm{z}\right|+3\right)\left(\left|\mathrm{z}\right|-1\right)}{\left(\left|\mathrm{z}\right|+1\right)}}\ge 2^3$

$\Rightarrow$ $\frac{\left(\left|\mathrm{z}\right|+3\right)\left(\left|\mathrm{z}\right|-1\right)}{\left(\left|\mathrm{z}\right|+1\right)}\ge 3$

$\Rightarrow$ $\left(\left|\mathrm{z}\right|+3\right)\left(\left|\mathrm{z}\right|-1\right)\ge 3\left(\left|\mathrm{z}\right|+1\right)$

$\Rightarrow$ ${\left|z\right|}^2+2\left|z\right|-3\ge 3\left|z\right|+3$

$\Rightarrow$ ${\left|z\right|}^2+\left|z\right|-6\ge 0$

$\Rightarrow$ $\left(\left|z\right|-3\right)\left(\left|z\right|+2\right)\ge 0$

Therefore, $\left|z\right|=3$ or $\left|z\right|=-2$ (not possible).

Since $\left|z\right|$ can not be a negative value, thus $\left|z\right|=-2$ is not possible.

$$\begin{gathered} \left|z\right|-3\ge 0 \\ \Rightarrow \left|z\right|\ge 3 \\ \Rightarrow {\left|z\right|}_{min}=3 \end{gathered}$$

Hence option $\left(B\right)$ is the correct option.