JEE PYQ: Complex Numbers Question 5
Question 5 - 2021 (17 Mar Shift 2)
Let $S_1, S_2$ and $S_3$ be three sets defined as
$S_1 = {z \in \mathbb{C} : |z - 1| \le \sqrt{2}}$
$S_2 = {z \in \mathbb{C} : \text{Re}((1-i)z) \ge 1}$
$S_3 = {z \in \mathbb{C} : \text{Im}(z) \le 1}$
Then the set $S_1 \cap S_2 \cap S_3$
(1) is a singleton
(2) has exactly two elements
(3) has infinitely many elements
(4) has exactly three elements
Show Answer
Answer: (3)
Solution
$S_1$: disk centered at $(1,0)$ with radius $\sqrt{2}$. $S_2$: $\text{Re}((1-i)(x+iy)) = x + y \ge 1$. $S_3$: $y \le 1$. The intersection of these three regions is a non-degenerate region with infinitely many elements.
BETA
