If the domain of the function $f(x) = \frac{\sqrt{x^2-25}}{4-x^2} + \log_{10}(x^2+2x-15)$ is $(-\infty, \alpha) \cup [\beta, \infty)$,then $\alpha^2 + \beta^3$ is equal to:

Let $f$ be a function such that $f(x) = \sum_{r=1}^n [r + \cos(\frac{x}{r})]$,where $[.]$ denotes the greatest integer function and $x \in [0, \pi]$. Then the range of $f(x)$ is:

Let $D = \{x \in R : f(x) = \sqrt{\frac{x-|x|}{x-[x]}} \text{ is defined} \}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4+x^2}$. Then $D \cap C =$

Let the domain of the function $f(x) = \cos^{-1}\left(\frac{4x+5}{3x-7}\right)$ be $[\alpha, \beta]$ and the domain of $g(x) = \log_2\left(2-6\log_{27}(2x+5)\right)$ be $(\gamma, \delta)$. Then $|7(\alpha+\beta)+4(\gamma+\delta)|$ is equal to . . . . . . .

$\left\{x \in R: \frac{2 x-1}{x^3+4 x^2+3 x} \in R\right\}$ equals

Domain of the function $\log |x^2 - 9|$ is