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

Correct Option (3)

Given that x is a real number such that 0 < x < 1.

Statement I: (which represents x² < x)

• Consider the inequality x² < x.
• Rearranging, we get x² - x < 0.
• Factoring out x, we have x(x - 1) < 0.
• Since 0 < x < 1, x is a positive number.
• Also, for 0 < x < 1, (x - 1) is a negative number.
• The product of a positive number and a negative number is always negative.
• Therefore, x(x - 1) < 0 is true for all real numbers x between 0 and 1.
• Hence, Statement I is correct.

Statement II: (which represents x < √x)

• Consider the inequality x < √x.
• Since x is between 0 and 1, both x and √x are positive. We can square both sides without changing the inequality direction.
• Squaring both sides yields x² < x.
• This is the same inequality as Statement I, which has already been proven to be correct for 0 < x < 1.
• Alternatively, consider an example: if x = 0.25, then √x = √0.25 = 0.5. Here, 0.25 < 0.5, which means x < √x holds true.
• Hence, Statement II is correct.

Since both Statement I and Statement II are correct, the appropriate option is (3).

Incorrect Options:

• Option (1) "I only" is incorrect because Statement II is also correct.
• Option (2) "II only" is incorrect because Statement I is also correct.
• Option (4) "Neither I nor II" is incorrect because both statements are correct.