A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola
$${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$$ at the point $$\left( {{x_1},{y_1}} \right)$$. Then $$x_1^2 + 5y_1^2$$ is equal to :

The domain of the function
f(x) = $${\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)$$ is (– $$\infty $$, -a]$$ \cup $$[a, $$\infty $$). Then a is equal to :

The value of

$${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$ is :

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :