Question

Let $S_1,S_2$ and $S_3$ be three sets defined as
$S_1=\{z\in C:|z-1|\le \sqrt{2}\}$
$S_2=\{z\in C:\mathrm{Re}((1-i)z)\ge 1\}$
$S_3=\{z\in C:\mathrm{Im}(z)\le 1\}$
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 $|z-1|\le \sqrt{2},z$ lies on and inside the circle of radius $\sqrt{2}$ units and centre $(1,0)$.

For $S_2$
Let $z=x+iy$
Now, $(1-i)(z)=(1-i)(x+iy)$
$Re((1-i)z)=x+y$
$\Rightarrow x+y\ge 1$
$\Rightarrow S_1\cap S_2\cap S_3$ has infinity many elements