The number of distinct real value of $\lambda$ for which the vectors $-{\mathrm{\lambda }}^2\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}-{\mathrm{\lambda }}^2\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-{\mathrm{\lambda }}^2\overline{\mathrm{k}}$ are coplanar is

$\mathrm{If} -{\mathrm{\lambda }}^2\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}-{\mathrm{\lambda }}^2\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}+-{\mathrm{\lambda }}^2\overline{\mathrm{k}}$ are coplanar then $$\begin{gathered} \left|\begin{array}{ccc}-{\mathrm{\lambda }}^2& 1& 1\\ 1& -{\mathrm{\lambda }}^2& 1\\ 1& 1& -{\mathrm{\lambda }}^2\end{array}\right|=0 \\ {C}_1\Rightarrow C_1+C_2+C_3 \\ \Rightarrow \left|\begin{array}{ccc}2-{\mathrm{\lambda }}^2& 1& 1\\ 2-{\mathrm{\lambda }}^2& -{\mathrm{\lambda }}^2& 1\\ 2-{\mathrm{\lambda }}^2& 1& -{\mathrm{\lambda }}^2\end{array}\right|=0 \\ \Rightarrow 2-{\mathrm{\lambda }}^2\left|\begin{array}{ccc}1& 1& 1\\ 1& -{\mathrm{\lambda }}^2& 1\\ 1& 1& -{\mathrm{\lambda }}^2\end{array}\right|=0 \\ {\mathrm{R}}_2\Rightarrow {\mathrm{R}}_2-{\mathrm{R}}_1,{\mathrm{R}}_3\Rightarrow {\mathrm{R}}_3-{\mathrm{R}}_1 \\ \Rightarrow \left(2-{\mathrm{\lambda }}^2\right)\left|\begin{array}{ccc}1& 1& 1\\ 0& -{\mathrm{\lambda }}^2-1& 0\\ 0& 0& -{\mathrm{\lambda }}^2-1\end{array}\right|=0 \\ \Rightarrow {\left({\mathrm{\lambda }}^2+1\right)}^2\left({\mathrm{\lambda }}^2-2\right)=0\Rightarrow {\mathrm{\lambda }}^2=2\Rightarrow \mathrm{\lambda }\pm \sqrt{2} \end{gathered}$$

No of distinct real values of $\lambda =2$