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 _________.

Step-by-step solution

log 1 2 z + 11 z - 1 2 ≤ 2 ⇒ z + 11 z - 1 2 ≥ 1 2 , ∵ log a b ≤ c ⇒ b ≥ a c   i f   0 < a < 1 ⇒ 2 z + 22 ≥ z - 1 2 , ∵ z ≠ 1   &   z - 1 2 > 0 ⇒ 2 z + 22 ≥ z 2 + 1 - 2 z ⇒ z 2 - 4 z - 21 ≤ 0 ⇒ z - 7 z + 3 ≤ 0 ⇒ z ≤ 7 ∴ Largest value of z is 7