The sum of all values of $\mathrm{\alpha }$, for which the points whose position vectors are $\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}},2\widehat{\mathrm{i}}-3\widehat{\mathrm{j}}+4\widehat{\mathrm{k}},(\mathrm{\alpha }+1)\widehat{\mathrm{i}}+2\widehat{\mathrm{k}}\text{ and }9\widehat{\mathrm{i}}+(\mathrm{\alpha }-8)\widehat{\mathrm{j}}+6\widehat{\mathrm{k}}$ are coplanar, is equal to

$$\begin{gathered} \begin{array}{l}\overline{\mathrm{OA}}=\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\\ \overline{\mathrm{OB}}=2\widehat{\mathrm{i}}-3\widehat{\mathrm{j}}+4\widehat{\mathrm{k}}\\ \overline{\mathrm{OC}}=(\mathrm{\alpha }+1)\widehat{\mathrm{i}}+0\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\end{array} \\ \begin{array}{l}\overline{\mathrm{OD}}=9\widehat{\mathrm{i}}+(\mathrm{\alpha }-8)\widehat{\mathrm{j}}+6\widehat{\mathrm{k}}\\ \overline{\mathrm{AB}}=\overline{\mathrm{OB}}-\overline{\mathrm{OA}}\\ =\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+\widehat{\mathrm{k}}\\ \overline{\mathrm{AC}}=\mathrm{\alpha }\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}-\widehat{\mathrm{k}}\\ \overline{\mathrm{AD}}=8\widehat{\mathrm{i}}+(\mathrm{\alpha }-6)\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\end{array} \end{gathered}$$
four points are coplanar then
$\begin{array}{l}[\overline{\mathrm{AB}},\overline{\mathrm{AC}},\overline{\mathrm{AD}}]=0\\ \left|\begin{array}{ccc}1& -1& 1\\ \mathrm{\alpha }& 2& -1\\ 8& \mathrm{\alpha }-6& 3\end{array}\right|=0\\ \Rightarrow {\mathrm{\alpha }}^2-2\mathrm{\alpha }-8=0\\ (\mathrm{\alpha }-4)(\mathrm{\alpha }+2)=0\\ \mathrm{\alpha }=4\mathrm{\alpha }=-2\end{array}$
Sum of the values of $\mathrm{\alpha }$ = 2