49. 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
2. B. II only
3. C. Both I and II
4. D. Neither I nor II

Answer: A

Explanation

Given that x is a real number between 0 and 1, i.e., 0 < x x³
To check this, we can divide by x² (since x > 0, the inequality sign doesn’t change): 1 > x. This is true, as given in the problem statement (0 < x 0.125, the statement x² > x³ is correct for 0 < x √x
To check this, we can square both sides (since both x and √x are positive, the inequality sign doesn’t change): x² > x. Now, divide by x (since x > 0): x > 1. This contradicts the given condition that 0 < x < 1. Alternatively, consider an example: Let x = 0.25. √x = √0.25 = 0.5. Here, x (0.25) is NOT greater than √x (0.5). In fact, 0.25 √x is incorrect for 0 < x < 1.

Therefore, only Statement I is correct. This question tests fundamental properties of real numbers and inequalities, a basic concept in CSAT.