If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right) + \log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha, \beta]$,then $3\alpha + 10\beta$ is equal to :

The domain of the function $f(x) = \log_{3+x}(x^2 - 1)$ is

If $R-(\alpha, \beta)$ is the range of $\frac{x+3}{(x-1)(x+2)}$,then the sum of the intercepts of the line $\alpha x+\beta y+1=0$ on the coordinate axes is

The domain of $f(x) = \log \left[(2.5)^{3-x^2} - (0.4)^{x+9}\right]$ is

$A$ and $B$ are subsets of $R$. Every element $x$ of $A$ is mapped to an element of $B$ by the rule, $y(x) = \begin{cases} \frac{5x}{(x-3)(x+3)} & \text{if } x \neq -1 \\ -1 & \text{if } x = -1 \end{cases}$, then $A =$

Let $\rho$ be a relation defined on the set of natural numbers $N$, as $\rho = \{(x, y) \in N \times N: 2x + y = 41\}$. Then the domain $A$ and range $B$ are: