Two cast iron blocks $$P$$ and $$Q,$$ each of $$500$$ $$mm$$ length, have the same cross-sectional area. Block $$P$$ has rectangular cross-section of $$100\,\,mm \times 200\,\,mm.$$ Block $$Q$$ is of square cross-section. Both $$P$$ and $$Q$$ were cast under the same conditions with all their surfaces enclosed within the mould. The solidification time of a casting is proportional to the square of the ratio of its volume to its surface area. The ratio of solidification time of the block $$P$$ to that of the block $$Q$$ is _________________

At $$x=0,$$ the function is $$f\left( x \right) = \left| {\sin {{2\pi x} \over L}} \right|\left( { - \infty < x < \infty ,L > 0} \right)$$

The eigen values of the matrix are $$\left[ {\matrix{ 0 & 1 \cr { - 1} & 0 \cr } } \right]$$

The number of solutions of the simultaneous algebraic equations $$y=3x+3$$ and $$y=3x+5$$ is