The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are

The function $$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$ has its maxima at

The value of the function, $$f\left( x \right) = \mathop {Lim}\limits_{x \to 0} {{{x^3} + {x^2}} \over {2{x^3} - 7{x^2}}}\,\,\,$$ is

A hydraulic structure has four gates which operate independently. The probability of failure of each gate is $$0.2.$$ Given that gate $$1$$ has failed, the probability that both gates $$2$$ and $$3$$ will fail is