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

The composition of an oxy-acetylene flame for cutting of steel contain more oxygen.

Parting-off operation is carried out on a cylindrical work-piece of $$100mm$$ diameter. The groove width is $$2$$ $$mm$$ and an in feed of $$0.2$$ $$mm$$ per revolution is given at a maximum cutting speed of $$60$$ $$m/mm.$$ The specific cutting force for the material is $$800\,\,N/m{m^2}.$$

The tangential force on the tool is

A part can be machined in $$30$$ minutes on an engine lathe where as it can be machined in $$6$$ minutes on a turret lathe. However, it would cost additionally Rs. $$500$$ to tool up the turret lathe for the operation. If the hourly rate including labour and overhead is Rs. $$80$$ for the engine lathe and Rs. $$160$$ for the turret lathe, The minimum number of parts required to make the turret lathe more economical to use, for the operation