Matrices & Determinants
System of Equations — Consistency Analysis
nta_pyq_2024_jan
Grade 12
Question:
Consider the system of linear equations $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4\lambda^2 z=\mu^2+15$, where $\lambda,\mu\in\mathbb{R}$. Which one of the following statements is NOT correct?
The system has unique solution if $\lambda\neq\frac{1}{2}$ and $\mu\neq1,15$
The system is inconsistent if $\lambda=\frac{1}{2}$ and $\mu\neq1$
The system has infinite number of solutions if $\lambda=\frac{1}{2}$ and $\mu=15$
The system is consistent if $\lambda\neq\frac{1}{2}$
Step-by-Step Solution
Key Concept: Compute $\Delta=(2\lambda-1)^2$. $\Delta=0\Rightarrow\lambda=\frac{1}{2}$. When $\lambda=\frac{1}{2}$, analyze $\Delta_x=\Delta_y=\Delta_z=(\mu-15)(\mu-1)$. Infinite solutions when $\mu=1$ or $\mu=15$; inconsistent otherwise.
$\Delta=(2\lambda-1)^2$. For $\lambda=\frac{1}{2}$: $\Delta=0$. $\Delta_x=\Delta_y=\Delta_z=(\mu-15)(\mu-1)=0$ when $\mu=1$ or $\mu=15$ (infinite solutions). If $\mu\neq1,15$: inconsistent. Statement (2) claims inconsistent whenever $\mu\neq1$, but $\mu=15$ also gives infinite solutions — so (2) is NOT correct.
Correct Answer: 2