$$\oint {{{{z^2} - 4} \over {{z^2} + 4}}} dz\,\,$$ evaluated anticlockwise around the circular $$\left| {z - i} \right| = 2,$$ where $$i = \sqrt { - 1} $$, is

The equation $$\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr } } \right]$$ has

A matrix has eigen values $$-1$$ and $$-2.$$ The corresponding eigenvectors are $$\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]$$ and $$\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]$$ respectively. The matrix is

A function $$y = 5{x^2} + 10x\,\,$$ is defined over an open interval $$x=(1,2).$$ At least at one point in this interval, $${{dy} \over {dx}}$$ is exactly