Let $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr {2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|,x \in [0,\pi ]$$. Then the maximum value of f(x) is equal to ______________.

Let $$F:[3,5] \to R$$ be a twice differentiable function on (3, 5) such that

$$F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt} $$. If $$F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}$$, then $$\alpha$$ + $$\beta$$ is equal to _______________.

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $$\to$$ S
such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m . n $$\in$$ S is equal to _____________.

If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0$$, with y(0) = 0, then $$5y'\left( {{\pi \over 2}} \right)$$ is equal to ______________.