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:

Let one root of the quadratic equation in $x$ :

$$ \left(k^2-15 k+27\right) x^2+9(k-1) x+18=0 $$

be twice the other. Then the length of the latus rectum of the parabola $y^2=6 k x$ is equal to:

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-a x+2=0$. Let the sets $\left\{a \in \mathbb{R}: e_1\right.$ and $e_2$ are the eccentricities of hyperbolas $\}=(\alpha, \beta)$, and $\left\{a \in \mathbb{R}: e_1\right.$ and $e_2$ are the eccentricities of an ellipse and a hyperbola, respectively $\}=(\gamma, \infty)$.

Then $\alpha^2+\beta^2+\gamma^2$ is equal to:

Let the set of all values of $k \in \mathbb{R}$ such that the equation $z(\bar{z}+2+i)+k(2+3 i)=0, z \in \mathrm{C}$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha+\beta)$ is equal to: