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

Consider the quadratic equation $(c - 5)x^2 - 2cx + (c - 4) = 0$,where $c \ne 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and the other root lies in the interval $(2, 3)$. Then the number of elements in $S$ is

For what interval of $m$ do all roots of the quadratic equation $x^2 - 2mx + m^2 - 1 = 0$ lie between $-2$ and $4$?

$f(x)=ax^2-bx-a$ is a quadratic expression. If $K$ is the least real number such that $f(x) \leq K, \forall x \in R$,then

The smallest value of $k$,for which both the roots of the equation $x^2-8kx+16(k^2-k+1)=0$ are real,distinct and have values at least $4$,is

For what set of values of $K$ do both roots of the equation $4x^2 - 20Kx + (25K^2 + 15K - 66) = 0$ lie below $2$?