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If the determinant of the matrix with rows (a, a², 1+a³) and (b, b², 1+b³) is zero, and the vectors (1,a,a²), (1,b,b²), and (1,c,c²) are not coplanar, then what is the value of abc?

1. 0
2. 2
3. −1
4. 1

Correct answer: −1

Solution

The determinant being zero indicates that the rows are linearly dependent, which implies that the values of a and b must be related in a specific way. Given that the vectors are not coplanar, the only way for this to hold true while satisfying the conditions is if the product abc equals -1, indicating a specific relationship among the values.

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