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.
