If logTan300(2|Z|2+2|Z|−3|Z|+1)<−2 then

2|z|2+2|z|−3|z|+1>(13)−2 2|z|2−|z|−6>0 (2|z|+3)(|z|−2)>0|z| 2It’s not possible ∴|z|>2

If logTan300(2|Z|2+2|Z|−3|Z|+1)<−2 then

2|z|2+2|z|−3|z|+1>(13)−2 2|z|2−|z|−6>0 (2|z|+3)(|z|−2)>0|z| 2It’s not possible ∴|z|>2

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If $\log_{T a n 30^{0}}^{(\frac{2 {| Z |}^{2} + 2 | Z | - 3}{| Z | + 1})} < - 2$ then

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

$| Z | < 2$

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

$| Z | > 2$

answer is C.

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$\frac{2 {| z |}^{2} + 2 | z | - 3}{| z | + 1} > {(\frac{1}{\sqrt{3}})}^{- 2} 2 {| z |}^{2} - | z | - 6 > 0 \begin{array}{l} (2 | z | + 3) (| z | - 2) > 0 \\ | z | < - \frac{3}{2} o r | z | > 2 \end{array}$
It’s not possible
$∴ | z | > 2$

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If logTan300(2|Z|2+2|Z|−3|Z|+1)<−2 then

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If $\log_{T a n 30^{0}}^{(\frac{2 {| Z |}^{2} + 2 | Z | - 3}{| Z | + 1})} < - 2$ then