The integral $$\,\int_{{x_1}}^{{x_2}} {{x^2}dx\,\,} $$ with $${x_2} > {x_1} > 0$$ is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $${\rm I}$$ is the exact value of the integral obtained analytically and $$J$$ is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

The quadratic equation $${x^2} - 4x + 4 = 0$$ is to be solved numerically, starting with the initial guess $${x_0} = 3.$$ The Newton- Raphson method is applied once to get a new estimate and then the Secant method is applied once using in the initial guess and this new estimate. The estimated value of the root after the application of the Secant method is ________.

Consider the following differential equation
$$x\left( {y\,dx + x\,dy} \right)\cos \left( {{y \over x}} \right)$$
$$\,\,\,\,\,\,\,\,\,\, = y\left( {x\,dy - y\,dx} \right)\sin \left( {{y \over x}} \right)$$

Which of the following is the solution of the above equation ($$C$$ is an arbitrary constant)

Consider the following probability mass function (p.m.f) of a random variable $$X.$$ $$$p\left( {x,q} \right) = \left\{ {\matrix{ q & {if\,X = 0} \cr {1 - q} & {if\,X = 1} \cr 0 & {otherwise} \cr } } \right.$$$
$$q=0.4,$$ the variance of $$X$$ is _______.