In a given day in the rainy season, it may rain$$70$$% of the time . If it rains, chance that a village fair will make a loss on that day is $$80$$%. However, if it does not rain, chance that the fair will make a loss on that day is only $$10$$%. If the fair has not made a loss on a given day in the rainy season, what is the probability that it has not rained on that day?

If $$\,\phi = 2{x^3}{y^2}{z^4}$$ then $${\nabla ^2}\phi $$ is

Directional derivative of $$\phi = 2xz - {y^2}$$ at the point $$(1, 3, 2)$$ becomes maximum in the direction of

In an arc welding operation carried out with a power source maintained at $$40$$ volts and $$400$$ amperes, the consumable electrode melts and just fills the gap between the metal plates to be butt-welded. The heat transfer efficiency for the process is $$0.8,$$ melting efficiency is $$0.3$$ and the heat required to melt the electrode is $$20$$ $$J/m{m^3}$$. If the travel speed of the electrode is $$4$$ $$mm/s$$, the cross-sectional area, in $$m{m^2}$$, of the weld joint is ________________.