The set of the all values of $\lambda $ for which the system of linear equations:

$2x_1-2x_2+x_3=\lambda x_1$

$2x_1-3x_2+2x_3=\lambda x_2$

$-x_1+2x_2=\lambda x_3 $ has a non-trivial solution,

contains two elements

The correct answer is option (1) : contains two elements

The given system of equations is

$x_1-(2-\lambda )-2x_2+x_3=0$

$2x_1-(3+\lambda )x_2+2x_3=0$

$-x_1+2x_2- \lambda x_3= 0 $

Clearly, it is homogenous system of equations and will have non-trivial solutions, if

$\begin{vmatrix}2-\lambda  & -2 & 1\\2 & -(3+\lambda) & 2\\-1 & 2 & -\lambda \end{vmatrix}=0$

$⇒ \lambda^3 - \lambda^2 -5\lambda + 3= 0 $

$⇒ \lambda^2( \lambda - 1) + 2 \lambda ( \lambda - 1) - 3 ( \lambda -1) = 0 $

$⇒ ( \lambda -1) ( \lambda^2 + 2 \lambda-3) = 0 $

$⇒( \lambda -1) ( \lambda-1) ( \lambda +3) = 0 $

$⇒ \lambda=1, 1, 3.$

Hence, there are two values of $ \lambda .$