The number of solutions of sin⁷x + cos⁷x = 1, x$$\in$$ [0, 4$$\pi$$] is equal to

If the domain of the function $$f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}$$ is the interval ($$\alpha$$, $$\beta$$], then $$\alpha$$ + $$\beta$$ is equal to :

Let f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$$

If f is continuous at x = 0, then $$\alpha$$ is equal to :

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x² $$-$$ y² = 3. If L is also a tangent to the parabola y² = $$\alpha$$x, then $$\alpha$$ is equal to :