The system of equation s:

$x+ y +z = 5 $

$x+ 2y + 3z= 9 $

$x+ 3y + \lambda z= \mu $ has a unique solution, if

$\lambda ≠ 5 $

The correct answer is Option (3) → $\lambda ≠ 5 $

$Δ=\begin{vmatrix}\begin{bmatrix}1&1&1\\1&2&3\\1&3&λ\end{bmatrix}\end{vmatrix}≠0$ for unique solution

$|Δ|=\begin{vmatrix}1&1&1\\1&2&3\\1&3&λ\end{vmatrix}$

$⇒C_2→C_2-C_1,C_3→C_3-C_1$

$|Δ|=\begin{vmatrix}1&0&0\\1&1&2\\1&2&λ-1\end{vmatrix}$

so $|Δ|=1×(λ-1-4)≠0$

so $λ≠5$