Matrices & Determinants
Infinite Solutions Condition
nta_pyq_2025_apr
Grade 12
Question:
If the system of equations $x + 2y - 3z = 2$, $2x + \lambda y + 5z = 5$, $14x + 3y + \mu z = 33$ has infinitely many solutions, then $\lambda + \mu$ is equal to:
Step-by-Step Solution
Key Concept: Set $\Delta = 0$ to get one relation between $\lambda$ and $\mu$, then use $\Delta_2 = 0$ to find $\mu$ explicitly.
$\Delta = 0 \Rightarrow \lambda\mu + 42\lambda - 4\mu + 107 = 0$. $\Delta_2 = 0 \Rightarrow \mu = 13$. Back-substituting: $\lambda = -1$. $\lambda + \mu = -1 + 13 = 12$.
Correct Answer: 12