A cylinder of $$150mm$$ diameter & $$200mm$$ height is to be cast with out any riser. The cylinder is moulded entirely in the drag of a green sand flask & is top gated. The cope of the flask is $$200mm$$ height & the height of the metal during pouring is $$50mm$$ above the cope. A tapered sprue is employed & the gating ratio is $$1:1.5:2$$. The time taken (in seconds) to fill the casting cavity neglecting energy losses, if the in-gate area is $$400sq.mm$$

The value of the following determinant $$\left[ {\matrix{ 1 & 4 & 9 \cr 4 & 9 & {16} \cr 9 & {16} & {25} \cr } } \right]$$ is

The matrix $$\left[ {\matrix{ 1 & { - 4} \cr 1 & { - 5} \cr } } \right]$$ is an inverse of the matrix $$\left[ {\matrix{ 5 & { - 4} \cr 1 & { - 1} \cr } } \right]$$

For the following matrix $$\left[ {\matrix{ 1 & { - 1} \cr 2 & 3 \cr } } \right]$$ the number of positive characteristic roots is