For the function of a complex variable w = lnz (where w = u + jv and z = x + jy) the u = constant lines get mapped in the z-plane as

The value of the counter integral $$$\int\limits_{\left| {z - j} \right| = 2} {{1 \over {{z^2} + 4}}\,} dz\,\,in\,the\,positive\,sense\,is$$$

For the differential equation $${{{d^2}y} \over {d{x^2}}} + {k^2}y = 0,$$ the boundary conditions are
(i) $$y=0$$ for $$x=0$$ and
(ii) $$y=0$$ for $$x=a$$
The form of non-zero solution of $$y$$ (where $$m$$ varies over all integrals ) are

For the matrix $$\left[ {\matrix{ 4 & 2 \cr 2 & 4 \cr } } \right].$$ The eigen value corresponding to the eigen vector $$\left[ {\matrix{ {101} \cr {101} \cr } } \right]$$ is