Four cards are randomly selected from a pack of $$52$$ cards. If the first two cards are kings, what is the probability that the third card is a king?

Consider the following second order linear differential equation $${{{d^2}y} \over {d{x^2}}} = - 12{x^2} + 24x - 20$$
The boundary conditions are: at $$x=0, y=5$$ and at $$x=2, y=21$$
The value of $$y$$ at $$x=1$$ is

For step-size, $$\Delta x = 0.4,$$ the value of following integral using Simpson's $$1/3$$ rule is ______

In Newton-Raphson iterative method, the initial guess value $$\left( {{x_{ini}}} \right)$$ is considered as zero while finding the roots of the equation: $$\,f\left( x \right) = - 2 + 6x - 4{x^2} + 0.5{x^3}.\,\,\,$$ The correction, $$\Delta x,$$ to be added to $${{x_{ini}}}$$ in the first iteration is __________.