The vectors $\mathbf{a}=2{\lambda }^2\mathbf{i}+4\lambda \mathbf{j}+\mathbf{k}$ and $\mathbf{b}=7\mathbf{i}-2\mathbf{j}+\lambda \mathbf{k}$ make an obtuse angle whereas the angle between $\mathbf{b}$ and $\mathbf{k}$ is acute and less than $\pi /6,$ then domain of $\lambda$ is :

As angle between a and b is obtuse, $\mathbf{\text{ a. }}\mathbf{b}<0$

$\begin{array}{l}\Rightarrow \left(2{\lambda }^2\mathbf{i}+4\lambda \mathbf{j}+\mathbf{k}\right)\cdot (7\mathbf{i}-2\mathbf{j}+\lambda \mathbf{k})<0\\ \Rightarrow 14{\lambda }^2-8\lambda +\lambda <0\\ \Rightarrow \lambda (2\lambda -1)<0\\ \Rightarrow 0<\lambda <\frac{1}{2} ...\left(i\right)\end{array}$

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

$\mathbf{b}\cdot \mathbf{k}=\left|\mathbf{b}\right|\cdot \left|\mathbf{k}\right|\cos \theta$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\lambda =\sqrt{53+{\lambda }^2}\cdot 1\cdot \cos \theta \\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overrightarrow{F}=\left(24\sqrt{3}\right)\theta <\frac{\pi }{6}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\cos \theta >\cos \frac{\pi }{6}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\cos \theta >\frac{\sqrt{3}}{2}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{\lambda }{\sqrt{53+{\lambda }^2}}>\frac{\sqrt{3}}{2}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4{\lambda }^2-3\left(53+{\lambda }^2\right)>0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\lambda }^2>159\end{array}$

$\Rightarrow \lambda <-\sqrt{159}\text{ or }\lambda >\sqrt{159}$             …(ii)

From(i) and (ii), $\lambda =\varphi .$

$\therefore$ Domain of $\lambda$ is null set.