Equation $${e^x} - 1 = 0\,\,$$ is required to be solved using Newton's method with an initial guess $$\,\,{x_0} = - 1.\,\,$$ Then after one step of Newton's method estimate $${x_1}$$ of the solution will be given by

Given $$X(z) = {z \over {{{(z - a)}^2}}}$$ with |z| > a, the residue of $$X(z){z^{n - 1}}$$ at z = a for $$n \ge 0$$ will be

A differential equation $${{dx} \over {dt}} = {e^{ - 2t}}\,\,u\left( t \right)\,\,$$ has to be solved using trapezoidal rule of integration with a step size $$h=0.01$$ sec. Function $$u(t)$$ indicates a unit step function. If $$x(0)=0$$ then the value of $$x$$ at $$t=0.01$$ sec will be given by

$$X$$ is uniformly distributed random variable that take values between $$0$$ and $$1.$$ The value of $$E\left( {{X^3}} \right)$$ will be