For a real number $x$,if the minimum value of $f(x) = x^2 + 2bx + 2c^2$ is greater than the maximum value of $g(x) = -x^2 - 2cx + b^2$,then:

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then which of the following is true?

If the graph of $y = ax^2 + bx + c$ is as follows,where $\Delta ABC$ is a right-angled isosceles triangle with hypotenuse $AC = 4\sqrt{2} \text{ units}$,then the minimum value of $ax^2 + bx + c$ is:

If the maximum value of $2x - 7 - ax^2$ cannot exceed $20$,then the minimum value of $a$ is

The quadratic expression $(2x+1)^{2} - px + q \neq 0$ for any real $x$, if

The maximum and minimum values of $\frac{x^2 + x + 1}{x^2 - x + 1}$ are respectively ...........