If the minimum value of the quadratic expression $x^2+5x-2$ is $M$ and it occurs at $x=a$,then $\frac{M}{a}$ is equal to

Match the following: Consider the equation $x^2 + 2(a - 1)x + a + 5 = 0$. Match the real values of $a$ with the conditions on the roots of the given equation.

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

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

If $f(x) = x^2 - 2(4K - 1)x + g(K) > 0$ for all $x \in R$ and for $K \in (a, b)$. If $g(K) = 15K^2 - 2K - 7$,then:

If both the roots of the equation $x^2 - 6ax + 2 - 2a + 9a^2 = 0$ exceed $3$,then

Let $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$,where $x \in R$. If $b$ and $c$ are nonzero real numbers such that $\min f(x) > \max g(x)$,then $\left|\frac{c}{b}\right|$ lies in the interval: