- Home
- Standard 11
- Mathematics
4-1.Complex numbers
hard
Let a complex number $z ,| z | \neq 1$, 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 ............
A
$8$
B
$7$
C
$6$
D
$5$
(JEE MAIN-2021)
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 $7$
Standard 11
Mathematics
