The domain of the function $f(x) = \sqrt{\log_{10}\left(\frac{5x - x^2}{4}\right)}$ is:

The domain of the function $f(x) = \sqrt{\frac{1-|x|}{2-|x|}}$ 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 the real-valued function $f(x) = \frac{\sqrt{\log_{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}}$ is (where $[x]$ denotes the greatest integer function):

If $f(x) = \frac{2x-3}{(x-2)(x-3)}$ is a real-valued function, then the value that $f(x)$ does not take is:

Let $f:[0,3] \rightarrow A$ be defined by $f(x)=2x^3-15x^2+36x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $g(x)=\frac{x^{2025}}{x^{2025}+1}$. If both the functions are onto and $S =\{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}$,then $n(S)$ is equal to :