The local maximum value of the function $$f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$$, x > 0, is

If the value of the integral
$$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta } $$, where $$\alpha$$, $$\beta$$ $$\in$$ R, 5$$\alpha$$ + 6$$\beta$$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($$\alpha$$ + $$\beta$$)² is equal to :

Let y(x) be the solution of the differential equation

2x² dy + (e^y $$-$$ 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :

The domain of the function $${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$$ is :