JEE Main 2021 — Mathematics Algebra

$${e}^{(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\mathrm{log}}_{e}2)}\geq {\mathrm{log}}_{\sqrt{2}}|5\sqrt{7}+9i|$$e(∣∣z∣+1∣(∣z∣+3)(∣z∣−1)​loge​2)≥log2​​∣57​+9i∣

$$\Rightarrow {2}^{\frac{(|z|+3)(|z|-1)}{(|z|+1)}}\geq lo{g}_{\sqrt{2}}(16)$$⇒2(∣z∣+1)(∣z∣+3)(∣z∣−1)​≥log2​​(16)

$$\Rightarrow {2}^{\frac{(|z|+3)(|z|-1)}{(|z|+1)}}\geq {2}^{3}$$⇒2(∣z∣+1)(∣z∣+3)(∣z∣−1)​≥23

$$\frac{(|z|+3)(|z|-1)}{(|z|+1)}\geq 3$$(∣z∣+1)(∣z∣+3)(∣z∣−1)​≥3

$$(|z|+3)(|z|-1)\geq 3(|z|+1)$$(∣z∣+3)(∣z∣−1)≥3(∣z∣+1)

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

$$\Rightarrow {|z|}^{2}+|z|-6\geq 0$$⇒∣z∣2+∣z∣−6≥0

$$\Rightarrow (|z|-3)(|z|+2)\geq 0\Rightarrow |z|-3\geq 0$$⇒(∣z∣−3)(∣z∣+2)≥0⇒∣z∣−3≥0

$$\Rightarrow |z|\geq 3\Rightarrow {|z|}_{\mathrm{min}}=3$$⇒∣z∣≥3⇒∣z∣min​=3