Find the area enclosed by the loop in the curve 4y² = 4x² $$-$$ x³.

If $$2f(xy) = {(f(x))^x} + {(f(y))^x}$$ for all $$x,y \in R$$ and $$f(1) = a( \ne 1)$$. Then $$\sum\limits_{k = 1}^n {f(k) = } $$

Let f(x) = x $$-$$ 3, g(x) = 4 $$-$$ x. Then the set of values of x for which $$|f(x) + g(x)|\, < \,|f(x)| + |g(x)|$$ is true, is given by :

$$\left\{ {x \in R:{{2x - 1} \over {{x^3} + 4{x^2} + 3x}} \in R} \right\}$$ is equal to