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 :

(a) I only ✓ UPSC's answer
(b) II only
(c) Both I and II
(d) Neither I nor II

Why the answer is (a)

• Let x be a real number such that 0 < x < 1.

• For statement I: Since 0 < x < 1, multiplying x by itself yields a smaller value, so x² < 1 and x³ < x². Thus, x² > x³ is correct.

• For statement II: Since 0 < x < 1, the square root of x is greater than x (e.g., if x = 0.25, √x = 0.5, and 0.5 > 0.25). Thus, x > √x is incorrect.

• Therefore, only statement I is correct.

• The correct option is (a) I only.

Why the other options are wrong

(b) II only: Statement II is incorrect because for any x between 0 and 1, √x is greater than x, not less.
(c) Both I and II: Statement II is incorrect, so both statements cannot be correct.
(d) Neither I nor II: Statement I is correct, so it is not the case that neither statement is correct.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.