Question

Let a complex number z, |z| ≠ 1, satisfy `log_(1/sqrt(2))((|z| + 11)/(|z| - 1)^2) ≤ 2`. Then, the largest value of |z| is equal to ______.

Options

• 5

• 8

• 6

• 7

Answer 1

Let a complex number z, |z| ≠ 1, satisfy `log_(1/sqrt(2))((|z| + 11)/(|z| - 1)^2) ≤ 2`. Then, the largest value of |z| is equal to 7.

Explanation:

Given that |z| ≠ 1

`log_(1/sqrt(2))((|z| + 11)/(|z| - 1)^2) ≤ 2`

Here base of logarithm lies between 0 and 1

So, 

⇒ `(|z| + 11)/(|z| - 1)^2 ≥ (1/sqrt(2))^2`

⇒ `(|z| + 11)/(|z| - 1)^2 ≥ 1/2`

⇒ 2|z| + 22 ≥ (|z| – 1)²

⇒ 2|z| + 22 ≥ |z|² – 2|z| + 1

⇒ |z|² – 4|z| – 21 ≤ 0

⇒ (|z| – 7)(|z| + 3) ≤ 0

⇒ |z| – 7 ≤ 0

⇒ |z| ≤ 7

So, the largest value of |z| is 7.