Question

Let $S_1,S_2$ and $S_3$ be three sets defined as
$S_1=\text{{}z\in C:|z-1|\le \sqrt{2}\text{}}$
$S_2=\text{{}z\in C:\text{Re}\left((1-i)z\right)\ge 1\text{}}$
$S_3=\text{{}z\in C:\text{Im}\left(z\right)\le 1\text{}}$
Then the set $S_1\cap S_2\cap S_3$

• A. Has infinitely many elements
• B. Has exactly two elements
• C. Has exactly three elements
• D. Is a singleton

Answer 1

The correct option is A Has infinitely many elements

Let, $z=x+iy$
$S_1\equiv {(x-1)}^2+y^2\le 2\cdots \left(1\right)$
$S_2\equiv x+y\ge 1\cdots \left(2\right)$
$S_3\equiv y\le 1\cdots \left(3\right)$

$\therefore S_1\cap S_2\cap S_3$ Has infinitely many elements.