Matrices & Determinants
Unique/No/Infinite Solutions — Identifying False Statement
nta_pyq_2023_apr
Grade None
Question:
For the system $x+y+z=6$, $x+2y+\alpha z=10$, $x+3y+5z=\beta$, which one of the following is NOT true?
System has no solution for $\alpha=3,\ \beta=24$
System has a unique solution for $\alpha=-3,\ \beta=14$
System has infinitely many solutions for $\alpha=3,\ \beta=14$
System has a unique solution for $\alpha=3,\ \beta\neq14$
Step-by-Step Solution
Key Concept: $\Delta=6-2\alpha$. Unique solution requires $\Delta\neq0$, i.e., $\alpha\neq3$. For $\alpha=3$, $\Delta=0$ — no unique solution.
For $\alpha=3$, $\Delta=0$; unique solution is impossible regardless of $\beta$.
Correct Answer: 4