Matrices & Determinants
Symmetric System — NOT Correct Statement
nta_pyq_2023_jan
Grade 12
Question:
For the system $\alpha x+y+z=1$, $x+\alpha y+z=1$, $x+y+\alpha z=\beta$, which statement is NOT correct?
It has infinitely many solutions if α=2 and β=-1
It has no solution if α=-2 and β=1
x+y+z=3/4 if α=2 and β=1
It has infinitely many solutions if α=1 and β=1
Step-by-Step Solution
Key Concept: $\Delta=0$: $\alpha^3-3\alpha+2=0\Rightarrow(\alpha-1)^2(\alpha+2)=0\Rightarrow\alpha=1$ or $\alpha=-2$. For $\alpha=2$: $\Delta=4\neq0$, so system has unique solution, not infinitely many.
Statement (1) is NOT correct.
Correct Answer: 1