What is the maximum value of the expression $\frac{x}{x^2 - 5x + 9}$ for all real values of $x$?

If $a, b$ are real numbers and $\alpha$ is a real root of $x^2 + 6x + 12 + 3 \sin(a + b\alpha) = 0$,then the value of $\cos(a + b\alpha)$ for the least positive value of $a + b\alpha$ is

If the set of all $a \in R$,for which the equation $2x^2 + (a-5)x + (15-3a) = 0$ has no real root,is the interval $(\alpha, \beta)$,and $X = \{x \in Z : \alpha < x < \beta\}$,then $\sum_{x \in X} x^2$ is equal to

The roots of the quadratic equation $x^2 - 2\sqrt{3}x - 22 = 0$ are:

If the roots of the equations $ax^2 + 2bx + c = 0$ and $bx^2 - 2\sqrt{ac}x + b = 0$ are real,then:

If $a, b, c$ are distinct and the roots of $(b-c)x^2 + (c-a)x + (a-b) = 0$ are equal,then $a, b$ and $c$ are in