Question : For two distinct real numbers x and y, which of them is bigger? Statement I : x² < y < 1 Statement II : y < √x < 1
- (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
- (b) Select this option if the question can be answered using either statement alone
- (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone
- (d) Select this option if the question cannot be answered even using any of the statements ✓ UPSC's answer
Why the answer is (d)
• Statement I states x² < y < 1, which implies y is positive and less than 1, but x can be any real number such that x² < y (e.g., x = 0.5 or x = -0.5), so the relationship between x and y is not fixed.
• Statement II states y < √x < 1, which implies x is positive and less than 1, but y can be any real number less than √x (e.g., y = 0.1 or y = -0.5), so the relationship between x and y is not fixed.
• Combining both statements, we have x² < y < √x < 1. This implies x is positive and x² < √x, which means x < 1 (since x > 0). However, it does not definitively determine if x > y or x < y without specific values.
• For example, if x = 0.25, then √x = 0.5. Statement I requires x² < y, so 0.0625 < y. Statement II requires y < 0.5. Let y = 0.1. Then x (0.25) > y (0.1).
• In another case, let x = 0.81, so √x = 0.9. Statement I requires 0.6561 < y. Statement II requires y < 0.9. Let y = 0.8. Then x (0.81) > y (0.8). Let y = 0.85. Then x (0.81) < y (0.85).
• Since different valid values of x and y satisfying both statements yield different comparisons (x > y or x < y), the question cannot be answered even using both statements together.
Why the other options are wrong
- (a) Select this option if the question can be answered using one of these statements alone…
- Neither statement alone is sufficient to determine which number is bigger, so the condition for option (a) is not met.
- (b) Select this option if the question can be answered using either statement alone
- Neither statement alone is sufficient to determine which number is bigger, so the condition for option (b) is not met.
- (c) Select this option if the question can be answered using both the statements together…
- Even using both statements together, the relationship between x and y is not uniquely determined, so the condition for option (c) is not met.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.