QJEEM21Macf2a: JEE Main 2021 Mathematics - Class 11 JEEM21Macf2a Solution: The least value of

Question

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The least value of |z| where z is complex number which satisfies the inequality

\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1}

, is equal to :

Accepted Answer

Let | z | = t, t

\ge

0

{e^{{{(t + 3)(t - 1)} \over {t + 1}}{{\log }_e}2}} \ge {\log _{\sqrt 2 }}16 = 8

(

\because

t + 1 > 0)

{2^{{{(t + 3)(t - 1)} \over {t + 1}}}} \ge {2^3}

{{(t + 3)(t - 1)} \over {t + 1}} \ge 3

{t^2} + 2t - 3 \ge 3t + 3

{t^2} - t - 6 \ge 0

t \in ( - \infty , - 2) \cup [3,\infty )

But t

\ge

0

\therefore

t \in [3,\infty )

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