Let $S = \{ x \in R:x \ge 0$ and $2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0\} $ then $S:$ . . .

Let $S=\{1,2,3, \ldots \ldots, n\}$ and $A=\{(a, b) \mid 1 \leq$ $a, b \leq n\}=S \times S$. A subset $B$ of $A$ is said to be a good subset if $(x, x) \in B$ for every $x \in S$. Then, the number of good subsets of $A$ is

Let $A=\{n \in N: H . C . F .(n, 45)=1\}$ and Let $B=\{2 k: k \in\{1,2, \ldots, 100\}\}$. Then the sum of all the elements of $A \cap B$ is

The number of elements in the set $\left\{n \in Z :\left|n^2-10 n+19\right| < 6\right\}$ is $...........$

If $A = \{x, y\}$ then the power set of $A$ is

Let the set $C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}$. Then $\sum_{(x, y) \in C}(x+y)$ is equal to