JEE Main 2021 (Online) 16th March Evening ShiftQuestion 1
The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp\left(\frac{(|z|+3)(|z|-1)}{||z|+1|}\log_e2\right)\ge\log_{\sqrt{2}}\left|5\sqrt{7}+9i\right|$, $i=\sqrt{-1}$, is equal to :
Complex Numbers
Solution
$|5\sqrt7+9i|=\sqrt{175+81}=16$, $\log_{\sqrt2}16=8$. With $r=|z|$: $2^{\frac{(r+3)(r-1)}{r+1}}\ge8\Rightarrow r^2+2r-3\ge3r+3\Rightarrow r^2-r-6\ge0\Rightarrow r\ge3$. Least $|z|=3$. Option (C).