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The homogeneous system 2x + 2ay + az = 0, 2x + 3by + bz = 0, 2x + 4cy + cz = 0 (with nonzero distinct real a, b, c) has a non-trivial solution. Then:

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

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

Solution

Setting the determinant to zero gives ab - 2ac + bc = 0, which on dividing by abc yields 1/a + 1/c = 2/b, i.e. 1/a, 1/b, 1/c in AP.

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