$\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ are two vectors such that }\left|\overrightarrow{a}\right|=1,\left|\overrightarrow{b}\right|=4\text{ and }\overrightarrow{a}\cdot \overrightarrow{b}=2\text{. If }\overrightarrow{c}=(2\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b}\text{, then the angle between }\overrightarrow{b}\text{ and }\overrightarrow{c}$

$\left|\overrightarrow{\mathrm{a}}\right|=1,\left|\overrightarrow{\mathrm{b}}\right|=4,\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}=2$

$\begin{array}{l}\overrightarrow{\mathrm{c}}=(2\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})-3\overrightarrow{\mathrm{b}}\\ \text{ or }\overrightarrow{\mathrm{c}}+3\overrightarrow{\mathrm{b}}=2\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\\ \because \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}=2\\ \Rightarrow \left|\overrightarrow{\mathrm{a}}\right|\cdot \left|\overrightarrow{\mathrm{b}}\right|\cos \mathrm{\theta }=2\\ \text{ or }\cos \mathrm{\theta }=\frac{2}{\left|\overrightarrow{\mathrm{a}}\right|\cdot \left|\overrightarrow{\mathrm{b}}\right|}=\frac{2}{4}\\ =\frac{1}{2}\\ \text{ or }\mathrm{\theta }=\frac{\mathrm{\pi }}{3}\\ \Rightarrow |\overrightarrow{\mathrm{c}}+3\overrightarrow{\mathrm{b}}{|}^2=|2\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}{|}^2\\ \text{ or }|\overrightarrow{\mathrm{c}}{|}^2+9|\overrightarrow{\mathrm{b}}{|}^2+2\overrightarrow{\mathrm{c}}\cdot 3\overrightarrow{\mathrm{b}}=4\left|\overrightarrow{\mathrm{a}}{|}^2\right|\overrightarrow{\mathrm{b}}{|}^2{\sin }^2\mathrm{\theta }\\ \text{ or }|\overrightarrow{\mathrm{c}}{|}^2+144+6\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=48\\ \text{ or }|\overrightarrow{\mathrm{c}}{|}^2+96+6(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})=0----\left(i\right)\end{array}$

$\text{ Now }\overrightarrow{c}=2\overrightarrow{a}\times \overrightarrow{b}-3\overrightarrow{b}$

$\begin{array}{l}|\overrightarrow{\mathrm{c}}{|}^2=4|\overrightarrow{\mathrm{a}}{|}^2|\overrightarrow{\mathrm{b}}{|}^2{\sin }^2\mathrm{\theta }+9|\overrightarrow{\mathrm{b}}{|}^2=192\\ \Rightarrow \overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=-48\end{array}$

$\text{ Again, putting the value of }\left|\overrightarrow{c}\right|\text{ in Eq. (i), we have }$

$\begin{array}{l}192+96+6\left|\overrightarrow{b}\right|\cdot \left|\overrightarrow{c}\right|\cos \alpha =0\\ \text{ or }6\times 4\times 8\sqrt{3}\cos \alpha =-288\end{array}$

$\begin{array}{l}\cos \mathrm{\alpha }=-\frac{288}{6\times 4\times 8\sqrt{3}}=-\frac{3}{2\sqrt{3}}\\ =-\frac{\sqrt{3}}{2}\\ \mathrm{\alpha }=\frac{5\mathrm{\pi }}{6}\end{array}$