Which of the following conditions implies that the roots of the equation $\frac{1}{4}x^2 + bx + c = 0$ are integers?

Which among the following equations has roots that are negatives of the roots of the equation $x^3-x^2+x-4=0$?

Assertion $(A)$: The maximum value of $-x^2+3x+1$ is $\frac{13}{4}$.
Reason $(R)$: If $a < 0$,the maximum value of $ax^2+bx+c$ exists at $x = -\frac{b}{2a}$.
The correct option among the following is

Let $f(x) = ax^{2} + bx + c$ be such that $f(1) = 3$,$f(-2) = \lambda$,and $f(3) = 4$. If $f(0) + f(1) + f(-2) + f(3) = 14$,then $\lambda$ is equal to...

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) $TRUE$?
$(A) (2, \frac{7}{2}, 6) \in S$
$(B) \text{If } (3, b, \frac{1}{12}) \in S, \text{ then } |2b| < 1$
$(C) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } ax + by = 1, bx + cy = -1 \text{ has a unique solution.}$
$(D) \text{For any given } (a, b, c) \in S, \text{ the system of linear equations } (a+1)x + by = 0, bx + (c+1)y = 0 \text{ has a unique solution.}$

If $x$ is a solution of the equation $\sqrt{2x + 1} - \sqrt{2x - 1} = 1$ for $x \ge \frac{1}{2}$,then $\sqrt{4x^2 - 1}$ is equal to: