Let a complex number z, |z| ≠ 1,
satisfy log _(1/√2)( (|z| + 11)/(|z| - 1)² ) ≤ 2. Then, the largest value of |z| is equal to ____________.

Solution

Maths sir · 3 steps

The base 1/√2 is less than 1, so the inequality flips the moment you remove the log.

1. Remove the log. Let r = |z|. The function log_(1/√2) is decreasing, so log_(1/√2) X ≤ 2 means X ≥ (1/√2)² = 1/2. The argument is always positive here, because r ≥ 0 and r ≠ 1.

   (r+11)/(r-1)² ≥ 1/2 ⇒ 2r + 22 ≥ r² - 2r + 1

2. Solve the quadratic. Since (r-1)² > 0, cross-multiplying is safe. Rearranging gives r² - 4r - 21 ≤ 0, that is (r-7)(r+3) ≤ 0, so -3 ≤ r ≤ 7.
3. So. With r ≥ 0 and r ≠ 1, the allowed values are 0 ≤ r ≤ 7 with r ≠ 1. The largest value of |z| is 7, option (d).

The idea to keep. When the base of a log lies between 0 and 1, the log is decreasing. Flip the inequality sign as you drop the log, then solve the plain algebraic inequality.

Answer from the NTA final answer key. Solution by PrepWiser.