"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 :"

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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

$\therefore$ $t \in [3,\infty )$"

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 :

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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 :

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