UPSC Prelims 2025 CSAT, question 79

Topic: Number system

Q79

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

I. $x^2>x^3$ .

II. $x>\sqrt{x}$ .

Select the correct answer using the code given below :

(a) I onlyCorrect answer
(b) II only
(c) Both I and II
(d) Neither I nor II

Solution.

Answer: (a) I only

Statement I is correct: for $0 < x < 1$ , multiplying the inequality $1 > x$ by $x^2 > 0$ gives $x^2 > x^3$ .
Statement II is incorrect: for $0 < x < 1$ , we have $0 < \sqrt{x} < 1$ , and multiplying $1 > \sqrt{x}$ by $\sqrt{x} > 0$ gives $\sqrt{x} > x$ (for example, at $x = 0.25$ , $\sqrt{x} = 0.5 > 0.25$ ), so $x > \sqrt{x}$ is false.

Trap: For any real number $x \in (0, 1)$ , taking a higher power makes the number smaller ( $x^3 < x^2 < x$ ), whereas taking the square root makes it larger ( $\sqrt{x} > x$ ).