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If the system of equations (λ − 1)x + (λ − 4)y + λz = 5 λx + (λ − 1)y + (λ − 4)z = 7 (λ + 1)x + (λ + 2)y − (λ + 2)z = 9 has infinitely many solutions, then λ^2 + λ is equal to

1. (1) 10
2. (2) 12
3. (3) 6
4. (4) 20

Correct answer: (2) 12

Solution

The system has infinitely many solutions when the determinant of the coefficient matrix is zero, leading to a condition on λ. Solving this condition reveals that λ^2 + λ equals 12, confirming option (2) as the correct answer.

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