The values of $λ$ for which the system of equation $x + 2y + z = 14,-x + y + z = 10, x + λy + z = 2$ has unique solution is

$R-\{2\}$; where R is set of real numbers.

The correct answer is Option (1) → $R-\{2\}$; where R is set of real numbers.

Coefficient matrix:

$\begin{bmatrix} 1 & 2 & 1 \\ -1 & 1 & 1 \\ 1 & \lambda & 1 \end{bmatrix}$

For a unique solution: $\det \ne 0$

Compute determinant:

$\det = 1\begin{vmatrix}1 & 1\\ \lambda & 1\end{vmatrix} -2\begin{vmatrix}-1 & 1\\ 1 & 1\end{vmatrix} +1\begin{vmatrix}-1 & 1\\ 1 & \lambda\end{vmatrix}$

$=1(1-\lambda)-2(-1-1)+(-\lambda-1)$

$=1-\lambda+4-\lambda-1$

$=4-2\lambda$

Unique solution condition:

$4 - 2\lambda \ne 0$

$\lambda \ne 2$

Answer: $\lambda \ne 2$