Let S1, S2 and S3 be three sets defined as


Let $S_{1}, S_{2}$ and $S_{3}$ be three sets defined as

$\mathrm{S}_{1}=\{\mathrm{z} \in \mathbb{C}:|\mathrm{z}-1| \leq \sqrt{2}\}$

$\mathrm{S}_{2}=\{\mathrm{z} \in \mathbb{C}: \operatorname{Re}((1-\mathrm{i}) \mathrm{z}) \geq 1\}$

$\mathrm{S}_{3}=\{\mathrm{z} \in \mathbb{C}: \operatorname{Im}(\mathrm{z}) \leq 1\}$

Then the set $S_{1} \cap S_{2} \cap S_{3}$


  1. (1) is a singleton

  2. (2) has exactly two elements

  3. (3) has infinitely many elements

  4. (4) has exactly three elements

Correct Option: , 3


For $|z-1| \leq \sqrt{2}, z$ lies on and inside the circle of radius $\sqrt{2}$ units and centre $(1,0)$.

For $\mathrm{S}_{2}$ Let $z=x+$ iy Now, $(1-i)(z)=(1-i)(x+i y)$

$\operatorname{Re}((1-i) z)=x+y$

$\Rightarrow x+y \geq 1$

$\Rightarrow S_{1} \cap S_{2} \cap S_{3}$ has infinity many elements


Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now