IIT-JEE 2008 · Paper 1, Mathematics question 21

The sets. B is the circle with centre $(2,1)$ and radius 3. For $z=x+iy$, $\operatorname{Re}((1-i)z)=x+y$, so C is the line $x+y=\sqrt2$. A is the half-plane $y\ge1$.

Line and circle. The distance from $(2,1)$ to the line is $\frac{3-\sqrt2}{\sqrt2}\approx1.12<3$, so they meet in two points. Solving, the points are approximately $(-0.76,\,2.18)$ and $(2.18,\,-0.76)$.

In A. Only the first has $y\ge1$.

(B) One element