The value of $x$ for which the angle between $\overline{a}=2x^2\overline{i}+4x\overline{j}+\overline{k}$ and $\overline{b}=7\overline{i}-2\overline{j}+x\overline{k}$ is obtuse and the angle between $\overline{b}$ and the z – axis is acute and less than $\frac{\pi }{6}.$

$\overline{a}.\overline{b}<0$

$14x^2-8x+x<0$

$7x(2x-1)<0\Rightarrow 0<x<\frac{1}{2}----1$

$\frac{(7,-2,x).(0,0,1)}{\sqrt{49+4+x^2}.1}>\frac{\sqrt{3}}{2}\Rightarrow \frac{x}{\sqrt{53+x^2}}>\frac{\sqrt{3}}{2}\Rightarrow x>0---2$

$\Rightarrow {\left(2x\right)}^2>3(53+x^2)\Rightarrow 4x^2>159+3x^2\Rightarrow x^2>159$

$x<-\sqrt{159}orx>\sqrt{159}----3$

There is no value satisfying equation 1, 2 and 3