Let $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function,
$$\phi \left( t \right) = {5 \over {12}} + t - {t^2}$$, then a² + b² is equal to :

If 1+(1–2².1)+(1–4².3)+(1-6².5)+......+(1-20².19)= $$\alpha $$ - 220$$\beta $$,
then an ordered pair $$\left( {\alpha ,\beta } \right)$$ is equal to:

Let x₀ be the point of Local maxima of $$f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$, where
$$\overrightarrow a = x\widehat i - 2\widehat j + 3\widehat k$$, $$\overrightarrow b = - 2\widehat i + x\widehat j - \widehat k$$, $$\overrightarrow c = 7\widehat i - 2\widehat j + x\widehat k$$. Then the value of
$$\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a $$ at x = x₀ is :

The integral $$\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx} $$ is equal to
(where C is a constant of integration):