If $X = \{ {8^n} - 7n - 1:n \in N\} $ and $Y = \{ 49(n - 1):n \in N\} ,$ then

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

The number of elements in the set $\left\{ n \in N : 10 \leq n \leq 100\right.$ and $3^{ n }-3$ is a multiple of $7\}$ is $........$.

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 $A=\left\{n \in N \mid n^{2} \leq n+10,000\right\}, B=\{3 k+1 \mid k \in N\}$ and $C=\{2 k \mid k \in N\}$, then the sum of all the elements of the set $A \cap(B-C)$ is equal to $.....$