Let $$P$$ be a variable point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ with foci $${F_1}$$ and $${F_2}$$. If $$A$$ is the area of the triangle $$P{F_1}{F_2}$$ then the maximum value of $$A$$ is ..........

Let $$C$$ be the curve $${y^3} - 3xy + 2 = 0$$. If $$H$$ is the set of points on the curve $$C$$ where the tangent is horizontal and $$V$$ is the set of the point on the curve $$C$$ where the tangent is vertical then $$H=$$.............. and $$V=$$ .................

The set of all $$x$$ for which $$in\left( {1 + x} \right) \le x$$ is equal to ..........

The larger of $$\cos \left( {In\,\,\theta } \right)$$ and $$In $$ $$\left( {\cos \,\,\theta } \right)$$ If $${e^{ - \pi /2}} < \theta < {\pi \over 2}$$ is ..................