Let $\overrightarrow{\mathrm{a}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+3\hat{\mathrm{k}},\overrightarrow{ \mathrm{b}}=3\hat{\mathrm{i}}-5\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{b}}$ and $(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}})·(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})=168$. Then the maximum value of $|\overrightarrow{\mathrm{c}}{|}^2$ is

$$\begin{gathered} \begin{array}{l}\overrightarrow{\mathrm{a}}=2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\\ \overrightarrow{\mathrm{b}}=3\widehat{\mathrm{i}}-5\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\\ \overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{b}}\\ \overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}=0\\ (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}=0\\ \Rightarrow \overrightarrow{\mathrm{c}}=\mathrm{\lambda }(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})\\ \overrightarrow{\mathrm{c}}=\mathrm{\lambda }(5\widehat{\mathrm{i}}-6\widehat{\mathrm{j}}+4\widehat{\mathrm{k}})\dots ..\left(1\right)\\ |\overrightarrow{\mathrm{c}}{|}^2={\mathrm{\lambda }}^2(25+36+16)\\ |\overrightarrow{\mathrm{c}}{|}^2=77{\mathrm{\lambda }}^2\\ (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}})\cdot (\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})=168\\ \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{b}}+|\overrightarrow{\mathrm{c}}{|}^2=168\end{array} \\ 14+\overrightarrow{\mathrm{c}}\cdot (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}})+77{\mathrm{\lambda }}^2=168 \end{gathered}$$
using equation (1)
$\begin{array}{l}\mathrm{\lambda }|5\widehat{\mathrm{i}}-6\widehat{\mathrm{j}}+4\widehat{\mathrm{k}}{|}^2+77{\mathrm{\lambda }}^2=154\\ 77\mathrm{\lambda }+77{\mathrm{\lambda }}^2-154=0\\ {\mathrm{\lambda }}^2+\mathrm{\lambda }-2=0\\ \mathrm{\lambda }=-2,1\end{array}$
$\therefore$ Maximum value of $|\overrightarrow{\mathrm{c}}{|}^2$ occurs when $\mathrm{\lambda }$ = –2
$|\stackrel{\rightharpoonup }{\mathrm{c}}{|}^2=77{\mathrm{\lambda }}^2$
= 77×4
= 308