Consider the following definite integral $$${\rm I} = \int\limits_0^1 {{{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}} \over {\sqrt {1 - {x^2}} }}dx} $$$
The value of the integral is

The divergence of the vector field $$\,V = {x^2}i + 2{y^3}j + {z^4}k\,\,$$ at $$x=1, y=2, z=3$$ is ________.

A two-faced fair coin has its faces designated as head $$(H)$$ and tail $$(T)$$. This coin is tossed three times in succession to record the following outcomes: $$H, H,H.$$ If the coin is tossed one more time, the probability (up to one decimal place ) of obtaining $$H$$ again, given the previous realizations of $$H, H$$ and $$H$$, would be _____.

The analysis of a water sample produces the following results: The total hardness (in mg/L as $${\mathrm{CaCO}}_3$$) of the water sample is ______.