The following algorithm computes the integral $$\,J = \int\limits_a^b {f\left( x \right)dx\,\,\,} $$ from the given values $${f_j} = f\left( {{x_j}} \right)$$ at equidistant points $$\,\,{x_0} = a,\,\,{x_1} = {x_0} + h,\,\,$$ $$\,{x_2} = {x_0} + 2h,...\,{x_{2m}} = {x_0} + 2mh = b\,\,$$ compute
$${S_0} = {f_0} + {f_{2m}}$$
$${S_1} = {f_1} + {f_3} + .... + {f_{2m - 1}}$$
$${S_2} = {f_2} + {f_4} + .... + {f_{2m - 2}}$$

$$J = {h \over 3}\left[ {{S_0} + 4\left( {{S_1}} \right) + 2\left( {{S_2}} \right)} \right]$$

The rule of numerical integration, which uses the above algorithm is

Which one of the following differential equations has a solution given by the function $$y = 5\sin \left( {3x + {\pi \over 3}} \right)$$

The solution of the differential equation $${{dy} \over {dx}} - {y^2} = 1$$ satisfying the condition $$y(0)=1$$ is

Two white and two black balls, kept in two bins, are arranged in four ways as shown below. In each arrangement, a bin has to be chosen randomly and only one ball needs to be picked randomly from the chosen bin. Which one of the following arrangements has the highest probability for getting a while ball picked?