While numerically solving the differential equation $$\,{{dy} \over {dx}} + 2x{y^2} = 0,y\left( 0 \right) = 1\,\,$$ using Euler's predictor corrector (improved Euler- Cauchy) method with a step size of $$0.2,$$ the value of $$y$$ after the first step is

It is known that two roots of the non-linear equation $$\,{x^3} - 6{x^2} + 11x - 6 = 0\,\,$$ are $$1$$ and $$3.$$ The third root will be

The polynomial $$\,p\left( x \right) = {x^5} + x + 2\,\,$$ has

Identity the Newton $$-$$ Raphson iteration scheme for the finding the square root of $$2$$