Question

Let $S _{1}, S _{2}$ and $S _{3}$ be three sets defined as
$S _{1}=\{z \in C :|z-1| \leq \sqrt{2}\}$
$S _{2}=\{ z \in C : \operatorname{Re}((1- i ) z ) \geq 1\}$
$S _{3}=\{ z \in C : \operatorname{Im}( z ) \leq 1\}$
Then the set $S_{1} \cap S_{2} \cap S_{3}$

• A. is a singleton
• B. has exactly two elements
• C. has infinitely many elements
• D. has exactly three elements

Answer 1

Answer: (C)

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 $z=x+i y$
Now, $(1-i)(z)=(1-i)(x+i y)$
${Re}((1- i ) z )= x + y$
$\Rightarrow x+y \geq 1$
$\Rightarrow S _{1} \cap S _{2} \cap S _{3}$ has infinity many elements