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.