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 _________.

The solution of the differential equation $$\,{x^2}{{dy} \over {dx}} + 2xy - x + 1 = 0\,\,\,$$ given that at $$x=1,$$ $$y=0$$ is

Using Cauchy's Integral Theorem, the value of the integral (integration being taken in counter clockwise direction)

$$\int\limits_C {{{{z^3} - 6} \over {3z - i}}} dz$$ is where C is |z| = 1

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.

The depth of neutral axis for the balanced failure is