Mathematics - Determinant Question with Solution | TestHub

Given system of equations is $$\begin{array}{rl}& x+y+z=6\\ & x+2y+3z=10\\ & x+2y+\lambda z=0\end{array}$$ It has unique solution. $$\begin{array}{rl}& \therefore \left|\begin{array}{lll}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|\ne 0\\ & \Rightarrow 1(2\lambda -6)-1(\lambda -3)+1(2-2)\ne 0\\ & \Rightarrow 2\lambda -6-\lambda +3\ne 0\Rightarrow \lambda -3\ne 0\Rightarrow \lambda \ne 3\end{array}$$