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# If $\overrightarrow a\:\:and\:\:\overrightarrow b$ are two unit vectors such that $\overrightarrow a+2\overrightarrow b\:\:and\:\:5\overrightarrow a-4\overrightarrow b$ are $\perp$ to each other, then the angle between $\overrightarrow a\:\:and\:\:\overrightarrow b$ is?

$\begin{array}{1 1} (A) \large\frac{\pi}{6}\\ (B) \large\frac{\pi}{4}\\ (C) \large\frac{\pi}{3} \\ (D) \cos^{-1}\large\frac{2}{7} \end{array}$

Given: $|\overrightarrow a|=|\overrightarrow b|=1\:\:and\:\:\overrightarrow a+2\overrightarrow b\:is\:\perp\:5\overrightarrow a-4\overrightarrow b$
$\Rightarrow\:(\overrightarrow a+2\overrightarrow b).(5\overrightarrow a-4\overrightarrow b)=0$
$\Rightarrow\:5|\overrightarrow a|^2+6(\overrightarrow a.\overrightarrow b)-8|\overrightarrow b|^2=0$
$\Rightarrow\:2cos\theta-1=0$ (where $\theta$ isthe angle between $\overrightarrow a\:\:and\:\:\overrightarrow b$.)
$cos\theta=\large\frac{1}{2}$$\:\:or\:\:\theta=\large\frac{\pi}{3}$