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Let p, q, r be nonzero reals which are respectively the 10th, 100th and 1000th terms of a harmonic progression. Consider the system: x + y + z = 1 10x + 100y + 1000z = 0 q r x + p r y + p q z = 0. Match List-I with List-II. List-I: (I) If q/r = 10; (II) If p/r != 100; (III) If p/q != 10; (IV) If p/q = 10. List-II: (P) x = 0, y = 10/9, z = -1/9 is a solution; (Q) x = 10/9, y = -1/9, z = 0 is a solution; (R) infinitely many solutions; (S) no solution; (T) at least one solution. Choose the correct matching.

1. (I) -> (T); (II) -> (R); (III) -> (S); (IV) -> (T)
2. (I) -> (Q); (II) -> (S); (III) -> (S); (IV) -> (R)
3. (I) -> (Q); (II) -> (R); (III) -> (P); (IV) -> (R)
4. (I) -> (T); (II) -> (S); (III) -> (P); (IV) -> (T)

Correct answer: (I) -> (Q); (II) -> (R); (III) -> (P); (IV) -> (R)

Solution

Working through the HP relations and substituting the candidate triples shows (I)->(Q), (II)->(R), (III)->(P), (IV)->(R), which is the JEE Advanced 2017 official answer.

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