Question

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

Explanation for the correct option:

Finding number of elements in the set $S_1\cap S_2\cap S_3$:

The given set $S_1=\left\{z\in C:|z–1|\le \sqrt{2}\right\}$

Considering , $z=x+iy$

$\Rightarrow S_1\equiv {(x–1)}^2+y^2\le 2\dots \left(1\right)$

It is a circle having radius $2$ and Centre $(1,0)$

And given $S_2=\left\{z\in C:Re\left(\right(1–i\left)z\right)\ge 1\right\}$

$\Rightarrow S_2\equiv x+y\ge 1\dots \left(2\right)$

It is an equation of line passing $(1,0)$

Also, given $S_3=\left\{z\in C:Im\left(z\right)\le 1\right\}$

$S_3\equiv y\le 1\dots \left(3\right)$

It is a line passing through (0, .1)

Now drawing the figure of $\left(1\right),\left(2\right)\&\left(3\right)$

Therefore from figure $S_1\cap S_2\cap S_3$ has infinitely many elements.

Therefore, the correct answer is option (A).