For what possible values of $a$ does $6$ lie between the roots of the equation $x^2 + 2(a - 3)x + 9 = 0$?

The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

Number of integral values of $a$ for which both roots of the quadratic equation $x^2 - (2a + 3)x + a^2 + 3a = 0$ lie in the interval $(0, 4)$ is:

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+2ax+(3a+10) = 0$ such that $\alpha < 1 < \beta$. Then the set of all possible values of $a$ is:

If the roots of ${x^2} + x + a = 0$ exceed $a$,then

If $ax^2 + bx + c < 0$ for all $x \in R$ and the expressions $cx^2 + ax + b$ and $ax^2 + bx + c$ have their extreme values at the same point $x$,then for the expression $cx^2 + ax + b$: