Solve the Following Questions


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. is a singleton

  2. has exactly two elements

  3. has infinitely many elements

  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 $S_{2}$

Let $\mathrm{z}=\mathrm{x}+$ iy

Now, $(1-\mathrm{i})(\mathrm{z})=(1-\mathrm{i})(\mathrm{x}+\mathrm{i} \mathrm{y})$

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

$\Rightarrow x+y \geq 1$

$\Rightarrow \mathrm{S}_{1} \cap \mathrm{S}_{2} \cap \mathrm{S}_{3}$ has infinity many elements

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