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UPSC CSE 2025 · PYQ · Number Properties / Inequalities · easy

Let x be a real number between 0 and 1. Which of the following statements is/are correct? I. x² > x³. II. x > √x. Select the correct answer using the code given below :

  1. A.I only✓ Correct
  2. B.II only
  3. C.Both I and II
  4. D.Neither I nor II

Explanation

For x between 0 and 1 (exclusive), powers of x decrease as the exponent increases. Statement I: x² > x³. For example, if x = 0.5, then x² = 0.25 and x³ = 0.125, so 0.25 > 0.125 ✓. This is true for all x in (0,1). Statement II: x > √x. For x = 0.5, √x ≈ 0.707, so x = 0.5 < 0.707. For numbers between 0 and 1, the square root is larger than the number itself, so x < √x, making Statement II false. Therefore only Statement I is correct.
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