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 satisfy the condition for consistency (rank(A) = rank(A|B) < number of variables).
The system is: <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> The coefficient matrix is A = [[1, 2, 3], [1, 3, 5], [2, 5, a]]. <br> For infinite solutions, det(A) = 0. <br> det(A) = 1(3a - 25) - 2(a - 10) + 3(5 - 6) = 3a - 25 - 2a + 20 - 3 = a - 8. <br> Setting det(A) = 0 gives a = 8. <br> Now check consistency for a = 8: <br> R<sub>2</sub> -> R<sub>2</sub> - R<sub>1</sub>: 0x<sub>1</sub> + x<sub>2</sub> + 2x<sub>3</sub> = 3 <br> R<sub>3</sub> -> R<sub>3</sub> - 2R<sub>1</sub>: 0x<sub>1</sub> + x<sub>2</sub> + 2x<sub>3</sub> = b - 12 <br> For infinite solutions, the equations must be consistent, so b - 12 = 3, which means b = 15.
Correct Answer: 3