The function to be maximized is given by $Z=3 x+2 y$. The feasible region for this function is the shaded region given below, then the linear constraints for this region are given by

If the line, $\frac{x-3}{2}=\frac{y+2}{1}=\frac{z+4}{3}$ lies in the plane, $\ell x+m y-z=9$, then $\ell^2+m^2$ is equal to

$$\int \frac{\sqrt{x}}{x+1} d x=$$

The equation of the normal to the curve $y=x \log x$, which is parallel to the line $2 x-2 y+3=0$, is