The necessary & sufficient condition for the differential equation of the form $$\,\,M\left( {x,y} \right)dx + N\left( {x,y} \right)dy = 0\,\,$$ to be exact is

The value of $$\varepsilon $$ in the mean value theoram of $$f\left( b \right) - f\left( a \right) = \left( {b - a} \right)\,\,f'\left( \varepsilon \right)$$ for $$f\left( x \right) = A{x^2} + Bx + C$$ in $$(a, b)$$ is

The void ratio of a soil sample is 1. The corresponding porosity of the sample is _______.

A prestressed concrete rectangular beam of size $$300\,\,mm \times 900\,\,mm$$ is prestressed with an intial prestressing force of $$700$$ $$kN$$ at an eccentricity of $$350$$ $$mm$$ at midspan. Stress at top due to prestress alone, in $$N/m{m^2}$$ is