The system of linear equations x + y + z = 6, x + 2y + 3z = 14 and 2x + 5y + pz = q have -
(A) infinitely many solution when p = 8, q = 36
(B) unique solution when p ≠ 8, q ≠ 36
(C) no solution when p = 8, q ≠ 36
(D) atleast one solution for q = 36, p ∈ R
Step-by-Step Solution
Key Concept: The system of linear equations can be analyzed using the determinant of the coefficient matrix D. If D \neq 0, there is a unique solution. If D = 0, the system may have infinitely many solutions or no solution depending on the consistency of the equations.
The coefficient matrix is A = [[1, 1, 1], [1, 2, 3], [2, 5, p]]. The determinant D = 1(2p - 15) - 1(p - 6) + 1(5 - 4) = 2p - 15 - p + 6 + 1 = p - 8. For a unique solution, D \neq 0, so p \neq 8. If p = 8, D = 0. The augmented matrix is [[1, 1, 1, 6], [1, 2, 3, 14], [2, 5, 8, q]]. Performing row operations: R2 -> R2 - R1 and R3 -> R3 - 2R1 gives [[1, 1, 1, 6], [0, 1, 2, 8], [0, 3, 6, q - 12]]. Then R3 -> R3 - 3R2 gives [[1, 1, 1, 6], [0, 1, 2, 8], [0, 0, 0, q - 36]]. For infinitely many solutions, we need q - 36 = 0, so q = 36. For no solution, we need q - 36 \neq 0, so q \neq 36.
Correct Answer: 1