Question

Let $A,B,C$ be three sets of complex numbers as defined below:
$A$ = {$z$ : Im $z\ge 1$ }
$B$ = {$z:\mid z-2-i\mid =3$}
$C$ = {$z:$Re$\left(\right(1-i\left)z\right)=\sqrt{2}$}
The number of elements in the set $A\cap B\cap C$ is

• A. 0
• B. 1
• C. 2
• D. $\infty$

Answer 1

The correct option is B 1

$1.\left)Im\right(z)>=1$

$y>=1-\left(i\right)$

=>A line parallel to x-axis with 1 unit distance including all point with magnitude greater than 1.

$2.)|z-2-i|=3$

This is a circle with center (2,1) and radius equal to 3 units on argand plane.z lies on the circle itself.

$3.\left)Re\right((1-i)z)=\sqrt{(}2)$

$=>x+y=\sqrt{(}2)$

Using the diagram to plot the above three curves,only 1 point falls in the shaded which is a common region to all of the curves i.e pt A