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If a, b, c are the roots of the cubic equation x³ - 5x² + 3x - 1 = 0, find the value of the determinant | a b c | | a-b b-c c-a | | b+c c+a a+b |
- 80
- 16
- 5
- 0
Correct answer: 80
Solution
Direct expansion of the determinant simplifies to a³ + b³ + c³ - 3abc. Using the standard identity this equals (a + b + c)(a² + b² + c² - ab - bc - ca). From Vieta's formulas for x³ - 5x² + 3x - 1: a + b + c = 5 and ab + bc + ca = 3, so a² + b² + c² = 25 - 6 = 19 and a² + b² + c² - ab - bc - ca = 19 - 3 = 16. The determinant is 5 * 16 = 80.
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