JEE Main 2021 (Online) 27th August Morning ShiftQuestion 1
If $A=\{x\in\mathbf{R}:|x-2|>1\}$, $B=\left\{x\in\mathbf{R}:\sqrt{x^{2}-3}>1\right\}$, $C=\{x\in\mathbf{R}:|x-4|\ge2\}$ and $\mathbf{Z}$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C)^{c}\cap\mathbf{Z}$ is $\_\_\_\_\_\_$ .
Sets, Relations and Function
Answer$256$
Solution
$A: x<1$ or $x>3$. $B: x^2>4\Rightarrow |x|>2$. $C: x\le2$ or $x\ge6$. $A\cap B\cap C=\{x<-2\}\cup\{x\ge6\}$. Complement $\cap\,\mathbf Z=\{-2,-1,0,1,2,3,4,5\}$, 8 elements. Subsets: $2^8=256$.