For a real number α, if the system <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[
Step-by-Step Solution
Key Concept: For a system of linear equations AX = B 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.
The system is AX = B where A = [[1, α, α^2], [α, 1, α], [α^2, α, 1]]. For infinitely many solutions, det(A) = 0. det(A) = 1(1 - α^2) - α(α - α^3) + α^2(α^2 - α^2) = 1 - α^2 - α^2 + α^4 = α^4 - 2α^2 + 1 = (α^2 - 1)^2 = 0. Thus α^2 = 1, so α = 1 or α = -1. If α = 1, the system becomes x + y + z = -1, x + y + z = 1, x + y + z = -1, which is inconsistent. If α = -1, the system becomes x - y + z = -1, -x + y - z = 1, x - y + z = -1, which simplifies to x - y + z = -1. This has infinitely many solutions. For α = -1, 1 + α + α^2 = 1 - 1 + 1 = 1.
Correct Answer: 1