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Greatest Integer Function

Let S denot...

Question

Let $S$ denote the set of complex numbers $z$ such that ${log}_{1 / 3} ({log}_{1 / 2} (| z |^{2} + 4 | z | + 3)) < 0$ , then $S$ is contained in

(0, 1)

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{z | R e (z) > 0}

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{z | R e (z) > 3}

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None of these

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Solution

The correct option is A None of these
${log}_{1 / 3} ({log}_{1 / 2} (| z |^{2} + 4 | z | + 3)) < 0$
$\Rightarrow {log}_{1 / 2} (| z |^{2} + 4 | z | + 3)) > (1 / 3)^{0} = 1$
$\Rightarrow | z |^{2} + 4 | z | + 3 < (1 / 2)^{1} = 1 / 2$
$\Rightarrow 2 | z |^{2} + 8 | z | + 5 < 0$
$\Rightarrow | z | \in (\frac{- 8 - \sqrt{64 - 40}}{4}, \frac{- 8 + \sqrt{64 - 40}}{4})$
But we know, $| z | \geq 0, \forall z \in C$
Hence, $z \in ϕ$

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Let S denote the set of complex numbers z such that log1/3(log1/2(|z|2+4|z|+3))<0, then S is contained in

The correct option is A None of theselog1/3(log1/2(|z|2+4|z|+3))<0⇒log1/2(|z|2+4|z|+3))>(1/3)0=1⇒|z|2+4|z|+3<(1/2)1=1/2⇒2|z|2+8|z|+5<0⇒|z|∈(−8−√64−404,−8+√64−404)But we know, |z|≥0,∀z∈CHence, z∈ϕ

Let $S$ denote the set of complex numbers $z$ such that ${log}_{1 / 3} ({log}_{1 / 2} (| z |^{2} + 4 | z | + 3)) < 0$ , then $S$ is contained in