$$\int\limits_0^\pi {\left[ {\cot x} \right]dx,} $$ where $$\left[ . \right]$$ denotes the greatest integer function, is equal to:

The area of the region bounded by the parabola $${\left( {y - 2} \right)^2} = x - 1,$$ the tangent of the parabola at the point $$(2, 3)$$ and the $$x$$-axis is :

Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if

$$\left| {\matrix{ a & {a + 1} & {a - 1} \cr { - b} & {b + 1} & {b - 1} \cr c & {c - 1} & {c + 1} \cr } } \right| + \left| {\matrix{ {a + 1} & {b + 1} & {c - 1} \cr {a - 1} & {b - 1} & {c + 1} \cr {{{\left( { - 1} \right)}^{n + 2}}a} & {{{\left( { - 1} \right)}^{n + 1}}b} & {{{\left( { - 1} \right)}^n}c} \cr } } \right| = 0$$

then the value of $$n$$ :

Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only
real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( { - 1} \right) < P\left( 1 \right),$$ then in the interval $$\left[ { - 1,1} \right]:$$