If the system of linear equations <br> x<sub>1</sub> + 2x<sub>2</sub> + 3x<sub>3</sub> = 6 <br> x<sub>1</sub> + 3x<sub>2</sub> + 5x<sub>3</sub> = 9 <br> 2x<sub>1</sub> + 5x<sub>2</sub> + ax<sub>3</sub> = b <br> is consistent and has infinite number of solutions, then :-
(A) a ∈ R - {8} and b ∈ R - {15}
(B) a = 8, b can be any real number
(C) a = 8, b = 15
(D) b = 15, a can be any real number
Step-by-Step Solution
Key Concept: For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the augmented matrix must have a rank less than the number of variables, which implies the consistency condition must be satisfied.
The system is consistent with infinite solutions if the determinant of the coefficient matrix is zero and the augmented matrix satisfies the consistency condition. The coefficient matrix is [[1, 2, 3], [1, 3, 5], [2, 5, a]]. Setting its determinant to zero: 1(3a - 25) - 2(a - 10) + 3(5 - 6) = 0 => 3a - 25 - 2a + 20 - 3 = 0 => a - 8 = 0 => a = 8. For consistency, the augmented matrix [1, 2, 3 | 6; 1, 3, 5 | 9; 2, 5, 8 | b] must have rank 2. Row operations: R2 -> R2 - R1 gives [1, 2, 3 | 6; 0, 1, 2 | 3; 2, 5, 8 | b]. R3 -> R3 - 2R1 gives [0, 1, 2 | b - 12]. Since R2 is [0, 1, 2 | 3], for consistency, b - 12 must equal 3, so b = 15.
Correct Answer: C