Let a complex number z,


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. 8

  2. 7

  3. 6

  4. 5

Correct Option:


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

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

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

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

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

$\Rightarrow|z| \leq 7$

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

Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now