$$\begin{gathered} \mathrm{Let} \stackrel{\rightharpoonup }{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{be} \mathrm{two} \mathrm{unit} \mathrm{vectors} \mathrm{such} \mathrm{that} \\ \left|\stackrel{\rightharpoonup }{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right|=\sqrt{3}. \mathrm{If} \stackrel{\rightharpoonup }{\mathrm{c}}=\overrightarrow{\mathrm{a}}+2\overrightarrow{\mathrm{b}}+3\left(\overrightarrow{\mathrm{a}}\times \stackrel{\rightharpoonup }{\mathrm{b}}\right), \\ \mathrm{then} 2\left|\overrightarrow{\mathrm{c}}\right| \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

$$\begin{gathered} \left|\overline{\mathrm{a}}\right|=1=\left|\overline{\mathrm{b}}\right| \\ \left|\overline{\mathrm{a}}+\overline{\mathrm{b}}\right|=\sqrt{3}\Rightarrow {\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}\right|}^2=3 \\ \Rightarrow 1+1+2\overline{\mathrm{a}}.\overline{\mathrm{b}}=3 \\ \Rightarrow \overline{\mathrm{a}}.\overline{\mathrm{b}}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ \overline{\mathrm{c}}=\overline{\mathrm{a}}+2\overline{\mathrm{b}}+3(\overline{\mathrm{a}}\times \overline{\mathrm{b}}) \\ {\left|\overline{\mathrm{c}}\right|}^2={\left|\overline{\mathrm{a}}\right|}^2+4{\left|\overline{\mathrm{b}}\right|}^2+9{\left|\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right|}^2+4 \overline{\mathrm{a}}.\overline{\mathrm{b}}+0+0 \\ =1+4+9({\left|\overline{\mathrm{a}}\right|}^2{\left|\overline{\mathrm{b}}\right|}^2-{\left(\overline{\mathrm{a}}.\overline{\mathrm{b}}\right)}^2)+4.\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ =5+9\left(1-\frac{1}{4}\right)+2 \\ =16-\frac{9}{4}=\frac{55}{4} \\ \left|\overline{\mathrm{c}}\right|=\frac{\sqrt{55}}{2} \end{gathered}$$