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If the system of linear equations 2x + 2ay + az = 0, 2x + 3by + bz = 0, 2x + 4cy + cz = 0 Where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then :

1. a, b, c are in G.P.
2. a, b, c are in A.P.
3. 1/a, 1/b, 1/c are in A.P.
4. a + b + c = 0

Correct answer: 1/a, 1/b, 1/c are in A.P.

Solution

For the system to have a non-zero solution, the determinant of the coefficients must be zero. This condition leads to the relationship among a, b, and c such that the reciprocals 1/a, 1/b, and 1/c form an arithmetic progression, indicating that their differences are constant.

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