If for the function $f(x) = \frac{1}{4}x^2 + bx + 10$; $f(12 - x) = f(x)$ for all $x \in R$,then the value of $b$ is:

Let $\alpha$ and $\beta$ be the roots of the equation $p x^2 + q x + r = 0$,where $p \neq 0$. If $p, q, r$ are in $AP$ and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$,then the value of $|\alpha - \beta|$ is

The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ assumes the least value is

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$ ($a \neq 0$; $a, b, c \in \mathbb{R}$),then $(1 + \alpha + \alpha^2)(1 + \beta + \beta^2)$ is . . . .

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-6x^2+11x-6=0$,then $\sum \alpha^2 \beta + \sum \alpha \beta^2 =$

If the difference between the roots of the equation $x^2 - px + 8 = 0$ is $2$,then $p = ......$