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

$$\begin{gathered} Given vectors are \\ \overline{a}=2x^2\overline{i}+4x\overline{j}+\overline{k},\overline{b} =7\overline{i}-2\overline{j}+x\overline{k} \\ Given that angle between \overline{a} and \overline{b} is obtuse then \\ \overline{a}.\overline{b}<0 \\ \Rightarrow 14{\mathrm{x}}^2-8\mathrm{x}+\mathrm{x}<0 \\ \Rightarrow 7\mathrm{x}(2\mathrm{x}-1)<0 \\ \Rightarrow 0<\mathrm{x}<1/2.............\left(\mathrm{i}\right) \\ Given angle between \overline{b} and z-axis is accute and less than \frac{\mathrm{\pi }}{6} \\ \Rightarrow (\overline{b},\overline{k})<\frac{\mathrm{\pi }}{6} \\ \Rightarrow \cos (\overline{b},\overline{k})>\cos \frac{\mathrm{\pi }}{6} \\ \Rightarrow \frac{\overline{b}.\overline{k}}{\left|\overline{b}\right|\left|\overline{k}\right|}>\frac{\sqrt{3}}{2} \\ \Rightarrow \frac{x}{\sqrt{49+4+x^2}}>\frac{\sqrt{3}}{2} \\ \Rightarrow 2x>\sqrt{3}\sqrt{53+x^2} \\ \Rightarrow 4{\mathrm{x}}^2>3{\mathrm{x}}^2+159 \\ \Rightarrow {\mathrm{x}}^2>159 \\ \Rightarrow \mathrm{x}>\sqrt{159} or \mathrm{x}<-\sqrt{159}.......\left(2\right) \end{gathered}$$

clearly, (i) and (ii) cannot hold together

$\therefore$there is no common value for equations (1) and (2)