The equation $\frac{3}{x - a^3} + \frac{5}{x - a^5} + \frac{7}{x - a^7} = 0$,where $a > 1$,has:

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 $72^x \cdot 48^y = 6^{xy}$,where $x$ and $y$ are non-zero rational numbers,then $x+y$ equals

Let $R^2$ denote $R \times R$. Let $S = \{(a, b, c) : a, b, c \in R \text{ and } ax^2 + 2bxy + cy^2 > 0 \text{ for all } (x, y) \in R^2 - \{(0, 0)\}\}$. Then which of the following statements is (are) $TRUE$?
$(A) (2, \frac{7}{2}, 6) \in S$
$(B) \text{If } (3, b, \frac{1}{12}) \in S, \text{ then } |2b| < 1$
$(C) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } ax + by = 1, bx + cy = -1 \text{ has a unique solution.}$
$(D) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } (a+1)x + by = 0, bx + (c+1)y = 0 \text{ has a unique solution.}$

Solve the equation $x^{2}+x+\frac{1}{\sqrt{2}}=0$.

If the roots of the equation $x^2 + 2x + p = 0$ are real,then what is the value of $p$?