Directions for the next 5 (five) items : Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly. Question : For two distinct real numbers x and y, which of them is bigger? Statement I : x² < y < 1 Statement II : y < √x < 1

Based on the foundational axioms of real analysis (formalized by mathematicians like Richard Dedekind in the late 19th century), solving algebraic inequalities requires evaluating all possible constraints to establish a definitive, unambiguous relationship.

To determine whether x > y or y > x, we evaluate the statements using test values for real numbers.

Statement I: x² y. However, y could also be 0.8, which means y > x. Because it yields multiple contradictory relationships, Statement I alone is insufficient.

Statement II: y y) or y could be 0.4 (y > x). Thus, Statement II alone is also insufficient.

Combining Both Statements: x² y) or y = 0.4 (resulting in y > x). Because a unique mathematical relationship cannot be established even when combining all data, Option D is the correct answer.

• Option A is incorrect because Statement I allows for scenarios where x is larger or smaller than y.
• Option B is incorrect because neither statement independently provides a unique solution.
• Option C is incorrect because synthesizing both statements still leaves an open interval around x, failing to restrict y to just one side of x.

Takeaway: For proper fractions/decimals between 0 and 1, higher powers yield smaller values (x² x). When a variable is bounded by a number's square and root, always test values both above and below the base number itself.