The least value of z where z is complex number which satisfies the inequality e z + 3 z - 1 z + 1 log e 2 ≥ log 2 5 7 + 9 i , i = - 1 , is equal to :

e z + 3 z - 1 z + 1 log e 2 ≥ log 2 5 7 + 9 i ⇒ 2 z + 3 z - 1 z + 1 ≥ l o g 2 16 ⇒ 2 z + 3 z - 1 z + 1 ≥ 2 3 z + 3 z - 1 z + 1 ≥ 3 z + 3 z - 1 ≥ 3 z + 1 z 2 + 2 z - 3 ≥ 3 z + 3 ⇒ z 2 + z - 6 ≥ 0 ⇒ z - 3 z + 2 ≥ 0 ⇒ z - 3 ≥ 0 ⇒ z ≥ 3 ⇒ z min = 3

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MathematicsComplex NumberGeneral(Modulus,Argument,Conjugate)Medium2 minPYQ_2021

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

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Answer:

Solution:

$e^{(\frac{(|z| + 3) (|z| - 1)}{||z| + 1|} \log_{e} 2)} \geq \log_{\sqrt{2}} |5 \sqrt{7} + 9 i|$

$\Rightarrow 2^{\frac{(|z| + 3) (|z| - 1)}{(|z| + 1)}} \geq l o g_{\sqrt{2}} (16)$

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

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

$(|z| + 3) (|z| - 1) \geq 3 (|z| + 1)$

${|z|}^{2} + 2 |z| - 3 \geq 3 |z| + 3$

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

$\Rightarrow (|z| - 3) (|z| + 2) \geq 0 \Rightarrow |z| - 3 \geq 0$

$\Rightarrow |z| \geq 3 \Rightarrow {|z|}_{\min} = 3$

Stream:JEESubject:MathematicsTopic:Complex NumberSubtopic:General(Modulus,Argument,Conjugate)

⏱ 2mℹ️ Source: PYQ_2021

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