The values of $a$ for which one root of the equation $x^2 - (a + 1)x + a^2 + a - 8 = 0$ exceeds $2$ and the other is lesser than $2$,are given by

If one root of $x^2+px-q^2=0$, where $p$ and $q$ are real, is less than $2$ and the other is greater than $2$, 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 both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$,then $m$ lies in the interval.

If $a$ and $b$ are the maximum and minimum values of the quadratic expressions $1-2x-5x^2$ and $x^2-2x+5$ respectively,then the set of all values of $x$ for which the expression $5ax^2+bx+7$ is positive,is

The number of integral values of $m$ for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$ is always positive for all $x \in R$ is: