The radius (in $cm$) of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone with radius $10 \ cm$ is

If a rectangle is inscribed in an equilateral triangle of side length $2 \sqrt{2}$ as shown in the figure,then the square of the largest area of such a rectangle is $....$

For the curve $y = x e^x$,the point:

The ordinates of the points on the curve $y = \tan^{-1}(\sin \sqrt{x})$,$0 \leq x \leq 8\pi^2$,at which the tangent is parallel to the $x$-axis are

The vertices of a triangle are $(0, 0)$,$(x, \cos x)$,and $(\sin^3 x, 0)$ where $0 < x < \frac{\pi}{2}$. The maximum area for such a triangle in sq. units is:

For all real values of $x,$ the minimum value of $\frac{1-x+x^{2}}{1+x+x^{2}}$ is