If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0, a \in R$ and $\alpha < 1 < \beta$,then:

$\frac{1}{2}(a+b+c)\{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}\}=?$

Let $f(x) = x^2 - x + k - 2$,$k \in R$. Then the complete set of values of $k$ for which $y = |f(|x|)|$ is non-derivable at $5$ distinct points is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $\frac{12}{\sqrt{x}} - \frac{23}{\sqrt{x}} = 5\sqrt{x}$
$II.$ $\frac{\sqrt{y}}{12} - \frac{5\sqrt{y}}{12} = \frac{1}{\sqrt{y}}$

Solve the given two equations and select the correct answer from the given options.
$I.$ $2.3x - 20.01 = 0$
$II.$ $2.9y - x = 0$

Solve the given two equations and choose the correct answer from the given options.
$I.$ $x^{2}+12x+36=0$
$II.$ $y^{2}+15y+56=0$