Matrices & Determinants
Determinants
nta_pyq_2025_jan
Grade 12
Question:
If the system $x+2y-3z=2,\ 2x+\lambda y+5z=5,\ 14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to:
Step-by-Step Solution
Key Concept: Two conditions $D_{2}=0$ and $D=0$ together pin $\mu$ and $\lambda.$
$D_{y}=\begin{vmatrix}1&2&-3\\2&5&5\\14&33&\mu\end{vmatrix}=0\Rightarrow\mu=13.$
$D=\begin{vmatrix}1&2&-3\\2&\lambda&5\\14&3&\mu\end{vmatrix}=0\Rightarrow \lambda\mu+42\lambda-4\mu+107=0.$
Substitute $\mu=13$: $13\lambda+42\lambda-52+107=0\Rightarrow 55\lambda+55=0\Rightarrow \lambda=-1.$
$\lambda+\mu=-1+13=12.$
Correct Answer: 3