Question

Let $A=\{z:|\arg (z+1)|\le \frac{\pi }{4}\},B=\{z:|z|\le 1\},C=\{z:|z-4|\le |z+2|\},$
then

• A. Area bounded by $A$ and $B$ is $\frac{\pi }{2}+1$
• B. Area bounded by $A$ and $B$ is $\frac{\pi }{4}+\frac{1}{2}$
• C. $A\cap B\cap C$ has exactly one element
• D. $A\cap B\cap C$ is an empty set

Answer 1

Answer: (A), (C)

The correct options are
A Area bounded by $A$ and $B$ is $\frac{\pi }{2}+1$
C $A\cap B\cap C$ has exactly one element


$A\cap B$ is the shaded region.
Area $=\text{Area of triangle+Area of circle}=2\times \frac{1}{2}\times 1\times 1+2\times \frac{\pi \left(1^2\right)}{4}=\frac{\pi }{2}+1$

$A\cap B\cap C=\left\{\right(1,0\left)\right\}$