For which of the following ordered pairs (μ, δ), the system of linear equations <br> x + 2y + 3z = 1 <br> 3x + 4y + 5z = μ <br> 4x + 4y + 4z = δ <br> is inconsistent?
Step-by-Step Solution
Key Concept: A system of linear equations is inconsistent if the determinant of the coefficient matrix is zero and the augmented matrix has a rank different from the coefficient matrix, or if the equations lead to a contradiction like 0 = k (where k is non-zero).
The system is given by the augmented matrix [1 2 3 | 1; 3 4 5 | \mu; 4 4 4 | \delta]. Performing row operations: R2 -> R2 - 3R1 and R3 -> R3 - 4R1 gives [1 2 3 | 1; 0 -2 -4 | \mu-3; 0 -4 -8 | \delta-4]. Further, R3 -> R3 - 2R2 gives [1 2 3 | 1; 0 -2 -4 | \mu-3; 0 0 0 | \delta-4 - 2(\mu-3)]. For the system to be inconsistent, the last row must represent a contradiction, i.e., 0 = \delta - 4 - 2\mu + 6, which simplifies to \delta - 2\mu + 2 \neq 0. Checking the options: (1) 0 - 2(1) + 2 = 0 (consistent), (2) 6 - 2(4) + 2 = 0 (consistent), (3) 4 - 2(3) + 2 = 0 (consistent), (4) 3 - 2(4) + 2 = -3 \neq 0 (inconsistent).
Correct Answer: 4