STD 11 - 14. probability — Maths STD 11 Science — Question

$ A=\{z: \operatorname{Im} z \geq 1\} $

$ B=\{z:|z-2-i|=3\} $

$ C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\} .$

$1.$  The number of elements in the set $\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}$ is

$(A)$ $0$ $(B)$ $1$ $(C)$ $2$ $(D)$ $\infty$

$2.$  Let $z$ be any point in $A \cap B \cap C$. Then, $|z+1-i|^2+|z-5-i|^2$ lies between

$(A)$ $25$ and $29$ $(B)$ $30$ and $34$ $(C)$ $35$ and $39$ $(D)$ $40$ and $44$

$3.$  Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $|w-2-i|<3$. Then, $|z|-|w|+3$ lies between

$(A)$ $-6$ and $3$ $(B)$ $-3$ and $6$

$(C)$ $-6$ and $6$ $(D)$ $-3$ and $9$

Give the answer question $1,2$ and $3.$