The system of equations:
$x+ y +z = 5 $
$x+ 2y + 3z= 9 $
$x+ 3y + \lambda z= \mu $ has a unique solution, if
Answer & explanation
Correct answer: option 3
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 $λ≠5$