The directional derivative of $$\,\,f\left( {x,y,z} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$ at the point $$P(2,1,3)$$ in the direction of the vector $${\mkern 1mu} \vec a = \widehat i - 2\widehat k{\mkern 1mu} $$ is _________.

For a given matrix $$A = \left[ {\matrix{ 2 & { - 2} & 3 \cr { - 2} & { - 1} & 6 \cr 1 & 2 & 0 \cr } } \right],$$ one of the eigen value is $$3.$$ The other two eigen values are

In the design of beams for the limit state of collapse in flexure as per $$IS:$$ $$456$$ - $$2000,$$ let the maximum strain in concrete be limited to $$0.0025$$ (in place of $$0.0035$$). For this situation, consider a rectangular beam section with breadth as $$250$$ $$mm,$$ effective depth as $$350$$ $$mm,$$ area of tension steel as $$1500\,\,m{m^2},$$ and characteristic strengths of concrete and steel as $$30$$ $$MPa$$ and $$250$$ $$MPa$$ respectively.

At the limiting state of collapse in flexure, the force acting on the compression zone of the section is

Assuming concrete below the neutral axis to be cracked, the shear stress across the depth of a singly-reinforced rectangular beam section