Matrices & Determinants
System of Equations — Consistency with Parameters
nta_pyq_2024_jan
Grade 12
Question:
Consider the system of linear equations $x+y+z=5$, $x+2y+\lambda^2 z=9$, $x+3y+\lambda z=\mu$, where $\lambda,\mu\in\mathbb{R}$. Then, which of the following statement is NOT correct?
System has infinite number of solution if $\lambda=1$ and $\mu=13$
System is inconsistent if $\lambda=1$ and $\mu\neq13$
System is consistent if $\lambda\neq1$ and $\mu=13$
System has unique solution if $\lambda\neq1$ and $\mu\neq13$
Step-by-Step Solution
Key Concept: Compute $\Delta$: set $\Delta=0\Rightarrow 2\lambda^2-\lambda-1=0\Rightarrow\lambda=1$ or $\lambda=-\frac{1}{2}$. Analyze consistency for each case. Note $\lambda=-\frac{1}{2}$ also makes $\Delta=0$ but is a separate case.
$\Delta=0\Rightarrow 2\lambda^2-\lambda-1=0\Rightarrow\lambda=1$ or $\lambda=-\frac{1}{2}$. For $\lambda=1$: $\Delta_z=0\Rightarrow\mu=13$ (infinite); $\mu\neq13$: inconsistent. For $\lambda=-\frac{1}{2}$: $\Delta=0$ but $\Delta_z\neq0$ (generally), so no solution. Statement (4) fails for $\lambda=-\frac{1}{2}$.
Correct Answer: 4