If the equation $\frac{1}{x} + \frac{1}{x - 1} + \frac{1}{x - 2} = 3x^3$ has $k$ real roots,then $k$ is equal to -

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

Given $f(x) = \frac{1}{2} - \tan^{-1}\left(\frac{\pi x}{2}\right)$ for $-1 < x < 1$ and $g(x) = \sqrt{3 + 4x - 4x^2}$. Find the domain of $(f + g)$.

The domain of the real valued function $f(x) = \frac{\sqrt{2-x} + \sqrt{1+x}}{\sqrt{x+3}}$ is

If $f(x)=\sqrt{2-x^2}$ and $g(x)=\ln (1-x)$ are two real-valued functions,then the domain of the function $(f+g)(x)$ is

If the domain of the function $\log _5(18 x-x^2-77)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)$ is $(\gamma, \delta)$,then $\alpha^2+\beta^2+\gamma^2$ is equal to :