$x + y + z = 6, \quad x + 2y + 3z = 10, \quad x + 2y + az = \beta$
Step-by-Step Solution
Key Concept: A system of linear equations has unique, infinite, or no solutions depending on the rank of coefficient matrix A and augmented matrix [A|B]. Here, |A| = a-3 determines consistency: when a≠3, unique solution exists by Cramer's rule; when a=3, the system reduces to two independent equations in three unknowns, requiring β=10 for consistency.
For the system with $|A| = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & a \end{vmatrix} = (a-3) \neq 0$, there is a unique solution when $a \neq 3$. When $a = 3$ and $\beta \neq 10$, there is no solution (inconsistent). When $a = 3$ and $\beta = 10$, there are infinitely many solutions. At least 2 solutions implies infinitely many solutions by the linear system theory.
Correct Answer: [A-q] [B-p, s] [C-p, r] [D-p, r]