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:

The value of $1^3-2^3+3^3-\ldots+15^3$ is:

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is: