The existence of a unique solution for the system of equations $x+y+z=\beta$,$5x-y+\alpha z=10$,and $2x+3y-z=6$ depends on:

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).

Solve the given two equations and select the correct answer from the given options.
$I.$ $15 x^{2}-41 x+14=0$
$II.$ $2 y^{2}-13 y+20=0$

$I. \quad p^{2}-18 p+77=0$
$II. \quad 3 q^{2}-25 q+28=0$
$Quantity \, 1$: Value of $p$
$Quantity \, 2$: Value of $q$

Sum of all distinct integral values of $\alpha$ such that the equation $x^2 - \alpha x + \alpha + 1 = 0$ has integral roots,is equal to-

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0$,where $a, b, c \in R$,have a common root,then $a:b:c = $ ...