The value of the sum $$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$ , where i = $$\sqrt { - 1} $$, equals

Let $${A_0}{A_1}{A_2}{A_3}{A_4}{A_5}$$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $${A_0}{A_1},{A_0}{A_2}$$ and $${A_0}{A_4}$$ is

If $$\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $$ then the value of $$f(1)$$ is

Suppose $$f(x)$$ is a function satisfying the following conditions
(a) $$f(0)=2,f(1)=1$$,
(b) $$f$$has a minimum value at $$x=5/2$$, and
(c) for all $$x$$, $$$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$$
where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.