Let a complex number z, |z| ≠ 1 , satisfy log 1 2 ( |z| + 11 (|z| -...

log\dfrac{1}{\sqrt{2}}\left(\dfrac{\left|z\right| + 11}{{\left(\left|z\right| - 1\right)}^{2}}\right) \le 2$</p><p>$⇒ \dfrac{\left|z\right| + 11}{{\left(\left|z\right| - 1\right)}^{2}} \ge \dfrac{1}{2}$, \left\{∵ \log_{a}\left(b\right) \le c ⇒ b \ge {a}^{c}if0 < a < 1\right\}$

$⇒ 2\left|z\right| + 22 \ge {\left(\left|z\right| - 1\right)}^{2}$, \left\{∵ \left|z\right| \ne 1 \text{ and } {\left(\left|z\right| - 1\right)}^{2} > 0\right\}$</p><p>$⇒ 2\left|z\right| + 22 \ge {\left|z\right|}^{2} + 1 - 2\left|z\right|$</p><p>$⇒ {\left|z\right|}^{2} - 4\left|z\right| - 21 \le 0$</p><p>$⇒ \left(\left|z\right| - 7\right)\left(\left|z\right| + 3\right) \le 0$</p><p>$⇒ \left|z\right| \le 7$</p><p>$∴$Largest value of \left|z\right|$ is $7$