Answer:
We know that vector \[\vec{a},\vec{b},\vec{c}\] are coplanar iff \[[\begin{matrix} {\vec{a}} & {\vec{b}} & {\vec{c}} \\ \end{matrix}]=0.\] It is given that \[\vec{a},\vec{b},\vec{c}\] are coplanar. \[\therefore \] \[[\begin{matrix} {\vec{a}} & {\vec{b}} & {\vec{c}} \\ \end{matrix}]=0\] \[\Rightarrow \] \[\left| \begin{matrix} 2 & -\,1 & 1 \\ 1 & 2 & -\,3 \\ 3 & \lambda & 5 \\ \end{matrix} \right|=0\] \[\Rightarrow \,\,2\,(10+3\lambda )+1\,(5+9)+(\lambda -6)=0\] \[\Rightarrow \] \[7\lambda +28=0\] \[\Rightarrow \] \[\lambda =-\,4\]
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