Matrices & Determinants
Consistency of System — Infinitely Many Solutions
nta_pyq_2026_jan
Grade 12
Question:
The system of linear equations
$x+y+z=6$
$2x+5y+az=36$
$x+2y+3z=b$
has
unique solution for $a=8$ and $b=16$
infinitely many solutions for $a=8$ and $b=16$
unique solution for $a=8$ and $b=14$
infinitely many solutions for $a=8$ and $b=14$
Step-by-Step Solution
Key Concept: Coefficient determinant $\Delta=8-a$. For $a=8$: $\Delta=0$, so unique solution is impossible. Check consistency: $R_2-2R_1$ gives $3y+(a-2)z=24$; $R_3-R_1$ gives $y+2z=b-6$. For $a=8$: $3y+6z=24\Rightarrow y+2z=8$. Consistent with $R_3-R_1$ only if $b-6=8$.
Infinitely many solutions for $a=8$ and $b=14$.
Correct Answer: 4