If $$A = \left( {1,\,1,\,1} \right),\,B = \left( {2,\, - 1,\,3} \right),\,C = \left( {0,\,4,\, - 2} \right),\,D = \left( {1,\,2,\,\lambda } \right)$$             and $$AB,\,AC$$   and $$AD$$  are coplanar then $$\lambda $$ is :

Solution :
Direction ratios of $$AB,\,AC$$   and $$AD$$  are respectively $$1,\, - 2,\,2;\, - 1,\,3,\, - 3;\,0,\,1,\,\lambda - 1.$$         They are coplanar $$ \Rightarrow $$  there is a line with direction ratios $$l,\,m,\,n$$   which is perpendicular to $$AB,\,AC$$   and $$AD.$$
$$\therefore \,l - 2m + 2n = 0,\, - l + 3m - 3n = 0,\,0.l + m + \left( {\lambda - 1} \right)n = 0$$
Eliminating $$l,\,m,\,n$$   we get \[\left| \begin{array}{l} \,\,\,\,\,\,1\,\,\, - 2\,\,\,\,\,\,\,\,\,2\\ - 1\,\,\,\,\,\,\,\,3\,\,\,\,\, - 3\\ \,\,\,\,\,\,0\,\,\,\,\,\,\,\,1\,\,\,\,\,\lambda - 1 \end{array} \right| = 0 \Rightarrow \lambda = 0.\]