A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.Question : Is $r^n$ less than 1, where $r$ is a real number and $n$ is a natural number ?Statement-I : $0 < r^2 < 1$Statement-II : $0 < r^3 < 2$Which 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 $r^n$ less than 1, where $r$ is a real number and $n$ is a natural number ?
Statement-I : $0 < r^2 < 1$
Statement-II : $0 < r^3 < 2$
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 by using one of the statements alone, but cannot be answered using the other statement alone

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 cannot be answered even by using both the Statements together

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

Statement I Sufficiency Analysis

Statement-I provides the inequality $0 < r^2 < 1$. This implies that the real number $r$ must lie strictly between $-1$ and $1$, and $r$ cannot be $0$. So, $-1 < r < 1$ and $r \neq 0$.

We need to determine if $r^n < 1$ is always true under this condition, where $n$ is a natural number.

Case 1: If $0 < r < 1$, raising $r$ to any natural power $n$ results in a value $r^n$ that is also between $0$ and $1$. Thus, $r^n < 1$.
Case 2: If $-1 < r < 0$, raising $r$ to an even power $n$ results in a positive value $r^n$ between $0$ and $1$. Raising $r$ to an odd power $n$ results in a negative value $r^n$. In both cases, $r^n < 1$.

Since $r^n < 1$ is true for all values of $r$ satisfying Statement-I and for all natural numbers $n$, Statement-I alone is sufficient to answer the question affirmatively.

Statement II Sufficiency Analysis

Statement-II provides the inequality $0 < r^3 < 2$. By taking the cube root, we find $0 < r < \sqrt[3]{2}$. Since $\sqrt[3]{2}$ is approximately $1.26$, this means $r$ is a positive number less than approximately $1.26$.

This condition alone does not guarantee that $r^n < 1$. Consider $r = 1.1$. This value satisfies $0 < r < \sqrt[3]{2}$ because $1.1^3 = 1.331$, which falls within the range $(0, 2)$.

However, if $n=1$, then $r^n = 1.1^1 = 1.1$. Since $1.1$ is not less than $1$, the condition $r^n < 1$ is not met.

Therefore, Statement-II alone is not sufficient to answer the question.

Combined Statements Analysis

To determine if both statements together are sufficient, we combine their conditions:

From Statement-I: $-1 < r < 1$ and $r \neq 0$.
From Statement-II: $0 < r < \sqrt[3]{2}$.
The intersection of these two ranges is $0 < r < 1$. This is because $1 < \sqrt[3]{2}$, so the interval $(0, 1)$ is contained within $(0, \sqrt[3]{2})$, and also satisfies $-1 < r < 1$.

If $0 < r < 1$, then for any natural number $n$, $r^n$ will always be positive and less than $1$. For example, $(0.5)^n$ is always less than $1$ for $n \ge 1$.

Thus, using both statements together, we can definitively answer that $r^n < 1$.

Conclusion

Statement-I alone is sufficient to answer the question ($r^n < 1$). Statement-II alone is not sufficient. Therefore, the question can be answered using one of the statements alone, but not using the other statement alone. This corresponds to Option A.

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