Consider the differential equation $${{{d^2}x\left( t \right)} \over {d{t^2}}} + 3{{dx\left( t \right)} \over {dt}} + 2x\left( t \right) = 0$$
Given $$x(0) = 20$$ & $$\,x\left( 1 \right) = {{10} \over e},$$ where $$e=2.718,$$

The value of $$x(2)$$ is

The Newton-Raphson method is used to solve the equation $$f\left( x \right) = {x^3} - 5{x^2} + 6x - 8 = 0.$$ Taking the initial guess as $$x=5$$, the solution obtained at the end of the first iteration is ________.

If $$C$$ is a circle of radius $$r$$ with centre $${z_0}$$ in the complex $$z$$-plane and if $$'n'$$ is a non-zero integer, then $$\oint\limits_c {{{dz} \over {{{\left( {z - {z_0}} \right)}^{n + 1}}}}} $$ equals

Consider the function $$g\left( t \right) = {e^{ - t}}\,\sin \left( {2\pi t} \right)u\left( t \right)$$ ,where $$u(t)$$ is the unit step function. The area under $$g(t)$$ is _______________.