$\text{ The values of }k\text{ for which the points }A(1,0,3),B(-1,3,4),C(1,2,1)\text{ and }D(k.2,5)\text{ are coplanar, are }$

$\text{ Let }a=(1,0,3),b=(-1,3,4),c=(1,2,1)\text{ and }d=(k,2,5)\text{ . Since }\mathrm{A},\mathrm{B},\mathrm{C}\text{ and }\mathrm{D}$

$\text{ are coplanar, we have }\left[dbc\right]+\left[dca\right]+\left[dab\right]=\left[abc\right]$

$\begin{array}{l}\Rightarrow \left|\begin{array}{c}k\\ -1\\ 1\end{array}\begin{array}{c} 2\\ 3\\ 2\end{array} \begin{array}{c}5\\ 4\\ 1\end{array}\right|+\left|\begin{array}{c}k \\ 1 \\ 1 \end{array}\begin{array}{c}2\\ 2\\ 0\end{array}\begin{array}{c} 5\\ 1\\ 3\end{array}\right|+\left|\begin{array}{ccc}k& 2& 5\\ 1& 0& 3\\ -1& 3& 4\end{array}\right|=\left|\begin{array}{c}1 \\ -1\\ 1\end{array}\begin{array}{c} 0\\ 3\\ 2\end{array}\begin{array}{c} 3\\ 4\\ 1\end{array}\right|\\ \Rightarrow -5k-15+6k-14-9k+1=-20\\ \Rightarrow -8k-28=-20\Rightarrow k=-1\end{array}$