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The determinant | a, b, a*α+b; b, c, b*α+c; a*α+b, b*α+c, 0 | equals zero if which of the following holds?

1. a, b, c are in AP
2. a, b, c are in GP
3. α is a root of the equation ax²+bx+c=0
4. (x-α) is a factor of ax²+2bx+c

Correct answer: a, b, c are in GP

Solution

The determinant reduces to -(b² - ac)(a*α² + 2b*α + c). It vanishes when b² = ac (a, b, c in GP) or when α is a root of ax² + 2bx + c = 0.

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