$\text{ If }\overline{a},\overline{b}\text{ are two vectors such that }\left|\overline{a}\right|=1,\left|\overline{b}\right|=4,\overline{a}·\overline{b}=2\text{ and }2\overline{a}\times \overline{b}-3\overline{b}=\overline{c}\text{ ,and the }$$angle between \overline{b},\overline{c} is x radians . The integral part of 2x is$

$\begin{array}{l}\stackrel{ }{\overline{b}·}\overline{c}=2\overline{a}\times \overline{b}·\overline{b}-3\overline{b}·\overline{b}=0-3b^2=-48,c^2=4(\overline{a}\times \overline{b}{)}^2+9b^2=4(16-4)+144=192\\ \left(\sin ce (\overline{a}\times \overline{b}{)}^2 ={\overline{a}}^2{\overline{b}}^2-{\left(\overline{a}·\overline{b}\right)}^2\right)\\ \Rightarrow \left|\overline{c}\right|=8\sqrt{3}\\ now \cos x=\frac{\overrightarrow{b}.\overrightarrow{c}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|}=\frac{-48}{4\times 8\sqrt{3}}=\frac{-\sqrt{3}}{2}\Rightarrow x=\frac{5\pi }{6}\Rightarrow 2x=\frac{5\pi }{3}\end{array}$