JEE PYQ: Complex Numbers Question 3
Question 3 - 2021 (16 Mar Shift 2)
The least value of $|z|$ where $z$ is complex number which satisfies the inequality
$$\exp\left(\frac{(|z|+3)(|z|-1)}{||z|+1|}\log_e 2\right) \ge \log_{\sqrt{2}}|5\sqrt{7} + 9i|$$
$i = \sqrt{-1}$, is equal to:
(1) $3$
(2) $\sqrt{5}$
(3) $2$
(4) $8$
Show Answer
Answer: (1)
Solution
RHS: $|5\sqrt{7} + 9i| = \sqrt{175 + 81} = 16$. $\log_{\sqrt{2}}16 = \frac{\log 16}{\log\sqrt{2}} = 8$. So $2^{\frac{(|z|+3)(|z|-1)}{|z|+1}} \ge 2^4$. Hence $(|z|+3)(|z|-1) \ge 4(|z|+1)$. Simplifying: $|z|^2 + |z| - 6 \ge 0 \Rightarrow |z| \ge 3$ (since $|z| \ge 0$). Minimum $= 3$.
BETA
