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If $\log_{\tan 30^{0}} (\frac{2 {|z|}^{2} + 2 |z| - 3}{|z| + 1}) < - 2$ then

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$|z| < \frac{3}{2}$

$|z| > \frac{3}{2}$

$|z| > 2$

$|z| < 2$

answer is C.

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Detailed Solution

$\frac{2 {|z|}^{2} + 2 |z| - 3}{|z| + 1} > 3$

$2 {|z|}^{2} + 2 |z| - 3 > 3 (|z| + 1)$

$2 {|z|}^{2} - |z| - 6 > 0$

$(|z| - 2) (2 |z| + 3) > 0$

$|z| > 2$

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