JEE PYQ: Complex Numbers Question 30
Question 30 - 2019 (09 Apr Shift 2)
Let $z \in \mathbb{C}$ be such that $|z| < 1$. If $\omega = \frac{5+3z}{5(1-z)}$, then:
(1) $5,\text{Re}(\omega) > 4$
(2) $4,\text{Im}(\omega) > 5$
(3) $5,\text{Re}(\omega) > 1$
(4) $5,\text{Im}(\omega) < 1$
Show Answer
Answer: (3)
Solution
$\omega = \frac{5+3z}{5-5z}$. With $|z| < 1$: $5\omega(1-z) = 5+3z \Rightarrow z = \frac{5\omega-5}{5\omega+3}$. $|z| < 1 \Rightarrow |5\omega-5| < |5\omega+3|$. Squaring and simplifying: $\text{Re}(\omega) > \frac{1}{5}$, i.e., $5,\text{Re}(\omega) > 1$.
BETA
