The graph of a quadratic equation $y = ax^2 + bx + c$ is a parabola that opens downwards and intersects the $x$-axis at two distinct points. Based on this description, which of the following is true?

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2 - x + \alpha = 0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1 - x_2| < 1$. Which of the following intervals is(are) a subset$(s)$ of $S$?
$(A) \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)$
$(B) \left(-\frac{1}{\sqrt{5}}, 0\right)$
$(C) \left(0, \frac{1}{\sqrt{5}}\right)$
$(D) \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$

If the sum of two particular roots of the equation $x^4-4x^3-7x^2+22x+24=0$ is equal to the sum of the remaining two roots,then the sum of the cubes of all the roots of this equation is

If $\alpha$ and $\beta$ are the least positive integers such that for all $n \in N$, $n^3+\alpha n$ is divisible by $3$ and $n^3-\beta n$ is divisible by $6$, then $\alpha+\beta=$

If $N$ denotes the number of quadratic equations with coefficients from the set $\{0, 1, 2, \dots, 9\}$ such that one root is $0$,then the value of $N$ is:

Let $p(x)$ be a quadratic polynomial with real coefficients. If $p(x)=0$ has only purely imaginary roots,then the zeroes of the polynomial $p(p(x))$ are