CDS 2026 · PYQ · Data Sufficiency / Exponents · medium
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is rⁿ less than 1, where r is a real number and n is a natural number?
Statement-I: 0 < r² < 1
Statement-II: 0 < r³ < 2
Which one of the following is correct in respect of the above Question and the Statements?
A.The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone✓ Correct
B.The Question can be answered by using either Statement alone
C.The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
D.The Question cannot be answered even by using both the Statements together
Explanation
From Statement-I: 0 < r² < 1 implies -1 < r < 1 and r ≠ 0. Here r could be negative (e.g., r = -0.5), then for odd n, rⁿ < 0 < 1; for even n, 0 < rⁿ < 1. In all cases rⁿ < 1, so Statement-I alone answers the question (yes). From Statement-II: 0 < r³ < 2 implies 0 < r < 2^(1/3) ≈ 1.26. If r > 1 (e.g., r = 1.2), then rⁿ can be greater than 1 for large n; if r < 1, then rⁿ < 1. So Statement-II alone is not sufficient. Therefore the question can be answered using one statement alone but not the other.
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