Question

Let a complex number $z,|z|\ne 1$, satisfy ${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1{)}^2}\right)\le 2$. Then, the largest value of $|z|$ is equal to

• A. 8
• B. 7
• C. 6
• D. 5

Answer 1

Answer: (B)

${\log }_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(Iz\mid -1{)}^2}\right)\le 2$
$\frac{|z|+11}{(|z|-1{)}^2}\ge \frac{1}{2}$
$2|z|+22\ge (|z|-1{)}^2$
$2|z|+22\ge |z{|}^2+1-2|z|$
$|z{|}^2-4|z|-21\le 0$
$\Rightarrow |z|\le 7$
$\therefore$ Largest value of $|z|$ is $7$