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If a, b, c are in A.P. and positive reals alpha, beta, gamma are in G.P., then the equation given by the determinant of rows (x + a, x² + log(alpha), k), (x + b, x² + log(beta), k), (x + c, x² + log(gamma), k) = 0 is:

1. is an identity
2. has a root x = 1
3. has a root x = 0
4. has real and identical roots

Correct answer: is an identity

Solution

Both column-1 entries (x+a, x+b, x+c) and column-2 entries (via the logs) are in A.P., so R1 + R3 - 2R2 makes a full row zero for every x, making the determinant identically zero.

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