QJEEM21Ma3a46: JEE Main 2021 Mathematics - Class 11 JEEM21Ma3a46 Solution: Let a complex number z,

Question

Verified Solution • 2025–26 Session moderate

Let a complex number z, |z|

\ne

1,

satisfy

{\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2

. Then, the largest value of |z| is equal to ____________.

Accepted Answer

{{|z| + 11} \over {{{(|z| - 1)}^2}}} \ge {1 \over 2}

2|z| + 22 \ge {(|z| - 1)^2}

2|z| + 22 \ge \,|z{|^2} - 2|z| + 1

|z{|^2} - 4|z| - 21 \le 0

(|z| - 7)(|z| + 3) \le 0

\Rightarrow \,|z| \le 7

\therefore

|z{|_{\max }} = 7

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