Forest ServicesQ49. 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:
Detailed Solution
Given that x is a real number between 0 and 1, i.e., 0 < x < 1. Statement I: x² > 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 < 1). Alternatively, consider an example: Let x = 0.5. x² = (0.5)² = 0.25 and x³ = (0.5)³ = 0.125. Since 0.25 > 0.125, the statement x² > x³ is correct for 0 < x < 1.
Statement II: 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 < 0.5.
So, the statement x > √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.
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