c
\(\begin{array}{l}
\left( {{\rm{\hat a}} + \;{\rm{2\hat b}}} \right)\, \bot \left( {5\hat a- 4\hat b} \right)\,\,\,\,\\ \therefore \left( {{\rm{\hat a}}\,\, + \;{\rm{2\hat b}}\,} \right)\,.\,\,\left( {\,5\hat a\,\, - 4\hat b\,} \right)\,\, = 0 \\ \Rightarrow 5\hat a.\,\hat a - 4\hat a.\,\hat b\,\, + \;10\hat b.\,\hat a - 8\hat b.\,\hat b\,\, = 0\\ \Rightarrow \,5+ \;\,6\hat a\,.\hat b\,\, - 8 = 0 \Rightarrow \,6\hat a\,.\,\hat b= 3\Rightarrow \hat a\,.\hat b = \frac{1}{2} \\ \Rightarrow \cos \theta  = \frac{1}{2} \Rightarrow \theta  = 60^\circ  \end{array}\)