The value of $$x$$ for which the matrix $$A = \left[ {\matrix{ 3 & 2 & 4 \cr 9 & 7 & {13} \cr { - 6} & { - 4} & { - 9 + x} \cr } } \right]$$ has zero as an eigen value is __________.

The matrix $$A = \left[ {\matrix{ a & 0 & 3 & 7 \cr 2 & 5 & 1 & 3 \cr 0 & 0 & 2 & 4 \cr 0 & 0 & 0 & b \cr } } \right]$$ has det
$$(A)=100$$ and trace $$(A)=14.$$ The value of $$\left| {a - b} \right|$$ is ___________.

As $$x$$ varies from $$- 1$$ to $$3,$$ which of the following describes the behavior of the function $$f\left( x \right) = {x^3} - 3{x^2} + 1?$$

Suppose $$C$$ is the closed curve defined as the circle $$\,\,{x^2} + {y^2} = 1\,\,$$ with $$C$$ oriented anti-clockwise. The value of $$\,\,\oint {\left( {x{y^2}dx + {x^2}ydx} \right)\,\,} $$ over the curve $$C$$ equals _______.