The vectors $\overline{\mathrm{a}}=2{\mathrm{\lambda }}^2\widehat{\mathrm{i}}+4\mathrm{\lambda }\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$ and  $\overline{\mathrm{b}}=7\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+\mathrm{\lambda }\widehat{\mathrm{k}}$ make an obtuse angle where as the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{k}}$ is acute and less than $\frac{\mathrm{\pi }}{6}$, then determine the domain of  $\mathrm{\lambda }$.

As angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ is obtuse,  $\overline{\mathrm{a}}.\overline{\mathrm{b}}<0$
$$\begin{gathered} \mathrm{As} \mathrm{angle} \mathrm{between} \overline{\mathrm{a}} \mathrm{and}\overline{ \mathrm{b}} \mathrm{is} \mathrm{obtuse}, \overline{\mathrm{a}} .\overline{ \mathrm{b}}<0 \\ \begin{array}{lc}\Rightarrow & \left(2{\mathrm{\lambda }}^2\widehat{\mathrm{i}}+4\widehat{\mathrm{\lambda }}\mathrm{j}+\widehat{\mathrm{k}}\right)\cdot (7\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+\mathrm{\lambda }\widehat{\mathrm{k}})<0\\ \Rightarrow & 14{\mathrm{\lambda }}^2-8\mathrm{\lambda }+\mathrm{\lambda }<0\\ \Rightarrow & \mathrm{\lambda }(2\mathrm{\lambda }-1)<0\\ \Rightarrow & 0<\mathrm{\lambda }<\frac{1}{2}\end{array} \end{gathered}$$

Angle between $\overline{\mathrm{b}}$ and  $\overline{\mathrm{k}}$ is acute and less than $\frac{\mathrm{\pi }}{6}$

$\begin{array}{lc} & \overline{\mathrm{b}}\cdot \overline{\mathrm{k}}=\left|\overline{\mathrm{b}}\right|\left|\overline{\mathrm{k}}\right|\cos \mathrm{\theta }\\ \Rightarrow & \mathrm{\lambda }=\sqrt{53+{\mathrm{\lambda }}^{2}1}\cos \mathrm{\theta }\\ \Rightarrow & \cos \mathrm{\theta }=\frac{\mathrm{\lambda }}{\sqrt{53+{\mathrm{\lambda }}^2}}\\ \Rightarrow & \mathrm{\theta }<\frac{\mathrm{\pi }}{6}\Rightarrow \cos \mathrm{\theta }>\cos \frac{\mathrm{\pi }}{6}\\ \Rightarrow & \cos \mathrm{\theta }>\frac{\sqrt{3}}{2}\Rightarrow \frac{\mathrm{\lambda }}{\sqrt{53+{\mathrm{\lambda }}^2}}>\frac{\sqrt{3}}{2}\\ \Rightarrow & 4{\mathrm{\lambda }}^2-3\left(53+{\mathrm{\lambda }}^2\right)>0\\ \Rightarrow & {\mathrm{\lambda }}^2>159\end{array}$