Let a complex number
Question:

Let a complex number $\mathrm{z},|\mathrm{z}| \neq 1$,

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

1. (1) 8

2. (2) 7

3. (3) 6

4. (4) 5

Correct Option: , 2

Solution:

$\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^{2}}\right) \leq 2$

$\frac{|z|+11}{(|z|-1)^{2}} \geq \frac{1}{2}$

$2|z|+22 \geq(|z|-1)^{2}$

$2|z|+22 \geq|z|^{2}+1-2|z|$

$|z|^{2}-4|z|-21 \leq 0$

$\Rightarrow|z| \leq 7$

$\therefore \quad$ Largest value of $|z|$ is 7