A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question: Is (0.5)ⁿ + (0.5)⁻ⁿ always greater than 2, where n is an integer?
Which one of the following is correct in respect of the above Question and the Statements?
- AThe Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
- BThe Question can be answered by using either Statement alone
- CThe Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
- DThe Question can be answered even without using both the Statements
Explanation
Taking n=0 (an integer) gives (0.5)⁰+(0.5)⁰=1+1=2, which is not greater than 2 — so the claim is false with no statements needed. (Indeed statement I, n even, includes n=0 giving 'No', and statement II, n negative, gives 2^k+2^{-k}>2, i.e. 'Yes'; each alone would also suffice.) Hence the question is answerable even without the statements, option D. Note: UPSC's published provisional key lists this question as A, but n=0 is a decisive counterexample, so it is flagged as disputed pending UPSC's final key.