Let [ • ] denote the greatest integer function. If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha, \beta)$, then $\alpha^2+\beta^2$ is equal to:

For the function $f:[1, \infty) \rightarrow[1, \infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements:

(I) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x)\right\}$ contains exactly two elements, and

(II) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x+1)\right\}$ is an empty set,

Let for some $\alpha \in \mathbb{R}, f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2 y^2+y+\alpha x y$ for all $x, y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum\limits_{n=1}^5(\alpha+f(n))$ is :

Let [•] denote the greatest integer function. If the domain of the function

$f(x)=\cos ^{-1}\left(\frac{4 x+2[x]}{3}\right)$ is $[\alpha, \beta]$, then $12(\alpha+\beta)$ is equal to :