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Let λ be a real number for which the system of linear equations x + y + z = 6, 4x + λy - λz = λ - 2, 3x + 2y - 4z = 5 has infinitely many solutions. Then λ is a root of the quadratic equation:
- λ² + λ - 6 = 0
- λ² - λ - 6 = 0
- λ² - 3λ - 4 = 0
- λ² + 3λ - 4 = 0
Correct answer: λ² - λ - 6 = 0
Solution
The correct option is right because for the system to have infinitely many solutions, the equations must be dependent, leading to a specific relationship between the coefficients of the variables, which results in the quadratic equation λ² - λ - 6 = 0.
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