The system of equations, given below, has $$$x+2y+4z=2$$$ $$$4x+3y+z=5$$$ $$$3x+2y+3z=1$$$

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

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

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?