During the filling process of a given sand mould cavity by metal through a horizontal runner of circular $$C.S,$$ the frictional head loss of the molten metal in the runner will increase with the

The value of $$q$$ for which the following set of linear equations $$2x+3y=0, 6x+qy=0$$ can have non-trivial solution is

If $$(1, 0, -1)$$$${}^T$$ is an eigen vector of the following matrix $$\left[ {\matrix{ 1 & { - 1} & 0 \cr { - 1} & 2 & { - 1} \cr 0 & { - 1} & 1 \cr } } \right]$$ then the corresponding eigen value is

If $$f\left( x \right) = \sin \left| x \right|\,\,$$ then the value of $${{df} \over {dx}}\,\,$$ at $$\,\,x = {{ - \pi } \over 4}\,\,$$ is