The error in numerically computing the integral $$\,\int\limits_0^\pi {\left( {\sin \,x + \cos \,x} \right)dx\,\,\,} $$ using the trapezoidal rule with three intervals of equal length between $$0$$ and $$\pi $$ is _________.

The value of $$\oint\limits_\Gamma {{{3z - 5} \over {\left( {z - 1} \right)\left( {z - 2} \right)}}dz} $$ along a closed path $$\Gamma $$ is equal to $$\left( {4\pi i} \right),$$ where $$z=x+iy$$ and $$i = \sqrt { - 1} .$$ The correct path $$\Gamma $$ is

A function $$f$$ of the complex variable $$z=x+iy,$$ is given as $$f(x,y)=u(x,y)+iv(x,y),$$
Where $$u(x,y)=2kxy$$ and $$v(x,y)$$ $$ = {x^2} - {y^2}.$$
The value of $$k,$$ for which the function is analytic, is __________.

A system of particles in motion has mass center $$G$$ as shown in the figure. The particle $$i$$ has mass $${m_i}$$ and its position with respect to a fixed point $$O$$ is given by the position vector $${r_i}$$ . The position of the particle with respect to $$G$$ is given by the vector $${\rho _i}$$ . The time rate of change of the angular momentum of the system of particles about $$G$$ is (The quantity $$\mathop {{\rho _i}}\limits^{..} $$ indicates second derivative of $${\rho _i}$$ with respect to time and likewise for $${r _i}$$).