The value of $$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} $$ is equal to:

Let a, b and c be the length of sides of a triangle ABC such that $${{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9}$$. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of $${R \over r}$$ is equal to :

Let f : N $$\to$$ R be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers x and y. If f(1) = 2, then the value of $$\alpha$$ for which

$$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)} $$

holds, is :