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

I. x² x³ .
II.

Question

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:

Accepted Answer

Statement I: x² > x³ Since x is a positive number (between 0 and 1), we can divide both sides by x³ without changing the inequality sign. x²/x³ > x³/x³ 1/x > 1 Given that 0 < x < 1, it is true that 1/x will always be greater than 1. For example, if x = 0.5, then 1/x = 2, which is greater than 1. Alternatively, consider x² - x³ = x²(1 - x). Since 0 < x < 1, x² is positive and (1 - x) is also positive. Therefore, x²(1 - x) is positive, meaning x² > x³. So, Statement I is correct. Statement II: x > √(x) Since x is positive, we can square both sides of the inequality without changing its direction: x² > x Rearrange the inequality: x² - x > 0 Factor out x: x(x - 1) > 0 Given that 0 < x < 1, x is a positive number. However, (x - 1) will be a negative number (e.g., if x = 0.5, then x - 1 = -0.5). A positive number multiplied by a negative number results in a negative number. Therefore, x(x - 1) < 0. This contradicts the inequality x(x - 1) > 0. So, Statement II is incorrect. (In fact, for 0 < x < 1, x < √(x). For example, if x = 0.25, then √(x) = 0.5, and 0.25 < 0.5). Since only Statement I is correct, the answer is A. The final answer is A

Answer

I only

Answer

II only

Answer

Both I and II

Answer

Neither I nor II

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:

Explanation

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