If $\mathrm{S}=\{\mathrm{a} \in \mathrm{R}:|2 \mathrm{a}-1|=3[\mathrm{a}]+2\{\mathrm{a}\}\}$, where $[\mathrm{t}]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to....................

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

Let $A =\{ x \in R :| x +1|<2\}$ and $B=\{x \in R:|x-1| \geq 2\}$. Then which one of the following statements is NOT true ?

Let $a>0, a \neq 1$. Then, the set $S$ of all positive real numbers $b$ satisfying $\left(1+a^2\right)\left(1+b^2\right)=4 a b$ is

If $\mathrm{A}=\{\mathrm{x} \in {R}:|\mathrm{x}-2|>1\}, \mathrm{B}=\left\{\mathrm{x} \in {R}: \sqrt{\mathrm{x}^{2}-3}>1\right\}$, $\mathrm{C}=\{\mathrm{x} \in {R}:|\mathrm{x}-4| \geq 2\}$ and ${Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^{c} \cap {Z}$ is .... .

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 $S:$ . . .