The square root of a number $$N$$ is to be obtained by applying the Newton $$-$$ Raphson iteration to the equation $$\,{x^2} - N = 0.\,\,$$ If $$i$$ denotes the iteration index, the correct iterative scheme will be

The solution of the differential equation $${{dy} \over {dx}} + {y \over x} = x$$ with the condition that $$y=1$$ at $$x=1$$ is

There are two containers with one containing $$4$$ red and $$3$$ green balls and the other containing $$3$$ blue balls and $$4$$ green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be ___________.

If $$\overrightarrow a $$ and $$\overrightarrow b $$ are two arbitrary vectors with magnitudes $$a$$ and $$b$$ respectively, $${\left| {\overrightarrow a \times \overrightarrow b } \right|^2}$$ will be equal to