Let x be a real number between 0 and 1. Which of the following statements is/are correct?

Answer: (a) I only

For a number between 0 and 1, higher powers get smaller and square roots get larger. So x² > x³ is true, but x > √x is false.

1. Take x = 0.25 as a test value.
2. I: x² = 0.0625 and x³ = 0.015625, so x² > x³. In general x² − x³ = x²(1 − x) > 0 for 0 < x < 1. True.
3. II: √0.25 = 0.5, which is larger than 0.25, so x > √x is false. For 0 < x < 1, √x is always larger than x.
4. Only statement I is correct.

• ✓ I x² − x³ = x²(1 − x) is positive when 0 < x < 1, so x² > x³.
• ✗ II For a fraction between 0 and 1 the square root is larger: √0.25 = 0.5 > 0.25.

Remember · Between 0 and 1: x³ < x² < x < √x. Test with x = 0.25 to confirm quickly.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). · Report an error