The quadratic expression $(2x+1)^{2} - px + q \neq 0$ for any real $x$, if

If both the roots of the equation $x^2-4ax+1-3a+4a^2=0$ exceed $1$,then $a$ lies in the interval

If $f:[1, 2] \rightarrow R$ defined by $f(x) = x^2 + 2kx + k$ is always negative for all $x \in [1, 2]$,then the interval in which $k$ lies is:

For the quadratic equation $x^2 - (K + 1)x + (K^2 + K - 8) = 0$,if one root is greater than $2$ and the other root is less than $2$,then $K$ lies in which interval?

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

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