If $$|\bar{a}|=2,|\bar{b}|=3,|\bar{c}|=5$$ and each of the angles between the vectors $$\bar{a}$$ and $$\bar{b}, \bar{b}$$ and $$\bar{c}$$, $$\bar{c}$$ and $$\bar{a}$$ is $$60^{\circ}$$, then the value of $$|\bar{a}+\bar{b}+\bar{c}|$$ is

The shaded region in the following figure represents the solution set for a certain linear programming problem. Then linear constraints for this region are given by

The function $\mathrm{f}$ defined on $$\left(-\frac{1}{3}, \frac{1}{3}\right)$$ by $$\mathrm{f}(x)=\left\{\begin{array}{cc} \frac{1}{x} \log \left(\frac{1+3 x}{1-2 x}\right) & , \quad x \neq 0 \\ \mathrm{k} & , \quad x=0 \end{array}\right.$$ is continuous at $$x=0$$, then $$\mathrm{k}$$ is

The mirror image of $$\mathrm{P}(2,4,-1)$$ in the plane $$x-y+2 z-2=0$$ is $$(\mathrm{a}, \mathrm{b}, \mathrm{c})$$, then the value of $$a+b+c$$ is