A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \le 1$$ is rotated about the $$x-$$axis by $$360$$ degrees. Area of the surface generated is

The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is

A six-face fair dice is rolled a large number of times. The mean value of the outcomes is _________.

The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1$$ and $${\left. {{{dy} \over {dx}}} \right|_{x = {\pi \over 2}}} = - 1$$ has