If $y = f(x) = ax^2 + 2bx + c = 0$ has imaginary roots and $4a + 4b + c < 0$,then :-

If non-zero real numbers $p$ and $q$ exist such that $\min f(x) > \max g(x)$,where $f(x) = x^2 + 2px + 2q^2$ and $g(x) = -x^2 - 2qx + p^2$ for $x \in \mathbb{R}$,find the set of values containing $|\frac{2p}{q}|$.

The number of values of $k,$ for which the equation $x^{2}-3x+k=0$ has two distinct roots lying in the interval $(0,1),$ 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:

If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

Match the following quadratic expressions with their minimum values:

Quadratic expression	The minimum value
i) $x^2 + 4x + 6$	a) $1$
ii) $x^2 - 2x + 5$	b) $2$
iii) $x^2 + 6x + 18$	c) $4$
iv) $x^2 - 4x + 5$	d) $9$