Matrices & Determinants
Unique/Infinite/No Solution — Identifying False Statement
nta_pyq_2023_apr
Grade 12
Question:
For the system $2x-y+3z=5$, $3x+2y-z=7$, $4x+5y+\alpha z=\beta$, which of the following is NOT correct?
The system has infinitely many solutions for $\alpha=-5$ and $\beta=9$
The system has infinitely many solutions for $\alpha=-6$ and $\beta=9$
The system is inconsistent for $\alpha=-5$ and $\beta=8$
The system has a unique solution for $\alpha\neq-5$ and $\beta=8$
Step-by-Step Solution
Key Concept: $\Delta=7\alpha+35$. Unique solution when $\Delta\neq0$, i.e., $\alpha\neq-5$. For $\alpha=-6$, $\Delta=-7\neq0$, so unique solution — not infinitely many.
$\Delta=7(-6)+35=-7\neq0$ for $\alpha=-6$. Unique solution, not infinite.
Correct Answer: 2