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 ?Statement-I : $n$ is an even integerStatement-II : $n$ is negativeWhich one of the following is correct in respect of the above Question and the Statements ?

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CDS - 2026

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 ?
Statement-I : $n$ is an even integer
Statement-II : $n$ is negative
Which one of the following is correct in respect of the above Question and the Statements ?

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CDS 2 2026 Maths Question Paper (13-Sep-2026)

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The Question can be answered even without using both the Statements

The Question can be answered by using either Statement alone

The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone

The correct answer is

The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone

To determine whether the expression $ (0.5)^n + (0.5)^{-n} $ is always greater than 2 for an integer $ n $, let's analyze the behavior under different conditions. Consider the expression:

$a = (0.5)^n + (0.5)^{-n}$

We must examine whether $ a > 2 $ under the given statements:

Statement I: $ n $ is an even integer
Statement II: $ n $ is negative

Let's analyze each statement:

Statement I: $ n $ is an even integer
- If $ n $ is an even integer, then $ n = 2k $ for some integer $ k $.
- Substituting, we have $ (0.5)^n = (0.5)^{2k} = (0.5^2)^k = (0.25)^k $ and $ (0.5)^{-n} = (0.5)^{-2k} = (2^2)^k = 4^k $.
- Thus, $ a = (0.25)^k + 4^k $.
- It's evident $ 4^k \geq 1 $ for $ k \geq 0 $, and $ (0.25)^k \leq 1 $, so $ a $ may not always be greater than 2 for non-negative even integers.
- For $ k < 0 $, $ (0.25)^k > 1 $, and $ 4^k < 1 $, so $ a = (0.25)^k + 4^k$ may or may not exceed 2 depending on $ k $.
Statement II: $ n $ is negative
- If $ n $ is negative, then $ (0.5)^n = 2^{|n|} $ and $ (0.5)^{-n} = 2^{-|n|} $.
- So, $ a = 2^{|n|} + 2^{-|n|} $.
- For $ n < 0 $, $ |n| > 0 $, thus $ 2^{|n|} \geq 1 $ and $ 2^{-|n|} < 1 $.
- Since $ f(x) = 2^x + 2^{-x} $ is always greater than 2 for any non-zero $ x $, $ a > 2 $ for all negative $ n $.

Based on our analysis:

Statement II is sufficient to determine that the expression is always greater than 2 for negative integers.
Statement I is not sufficient by itself because it doesn't ensure that the expression exceeds 2 for all even integers.

Hence, the correct answer is: The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.

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