The least value of |z| where ‘z’ is complex number which satisfies the in-equality exp((|z|+3)(|z|−1)(|z|+1)loge2)>log2|57+9i|, i=−1, is equal to:

2(|z|+3)(|z|−1)(|z|+1)≥23⇒(|z|+3)(|z|−1)|z|+1≥3⇒|z|2+2|z|−3≥3|z|+3⇒|z|2−|z|−6≥0 (|z|−3)(|z|+2)≥0

The least value of |z| where ‘z’ is complex number which satisfies the in-equality exp((|z|+3)(|z|−1)(|z|+1)loge2)>log2|57+9i|, i=−1, is equal to:

2(|z|+3)(|z|−1)(|z|+1)≥23⇒(|z|+3)(|z|−1)|z|+1≥3⇒|z|2+2|z|−3≥3|z|+3⇒|z|2−|z|−6≥0 (|z|−3)(|z|+2)≥0

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The least value of $| z |$ where ‘z’ is complex number which satisfies the in-equality $\exp (\frac{(| z | + 3) (| z | - 1)}{(| z | + 1)} \log_{e}^{2}) > \log_{\sqrt{2}} | 5 \sqrt{7} + 9 i |, i = \sqrt{- 1},$ is equal to:

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$\sqrt{5}$

answer is D.

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$2^{\frac{(| z | + 3) (| z | - 1)}{(| z | + 1)}} \geq 2^{3} \Rightarrow \frac{(| z | + 3) (| z | - 1)}{| z | + 1} \geq 3 \Rightarrow {| z |}^{2} + 2 | z | - 3 \geq 3 | z | + 3$

$\Rightarrow {| z |}^{2} - | z | - 6 \geq 0 (| z | - 3) (| z | + 2) \geq 0$

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The least value of |z| where ‘z’ is complex number which satisfies the in-equality exp((|z|+3)(|z|−1)(|z|+1)loge2)>log2|57+9i|, i=−1, is equal to:

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The least value of $| z |$ where ‘z’ is complex number which satisfies the in-equality $\exp (\frac{(| z | + 3) (| z | - 1)}{(| z | + 1)} \log_{e}^{2}) > \log_{\sqrt{2}} | 5 \sqrt{7} + 9 i |, i = \sqrt{- 1},$ is equal to: