Q. 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:
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
Correct Answer : (a) I only
UPSC Prelims 2025 CSAT
Explanation :
Let 0 < x < 1.
1. Check I: x² > x³
– Since x > 0, we can divide both sides of x² > x³ by x² to get
1 > x.
– And indeed 1 > x holds for all x in (0,1).
⇒ Statement I is true.
2. Check II: x > √x
– Since x > 0, divide both sides of x > √x by √x to get
√x > 1.
– That is equivalent to x > 1, which contradicts x < 1.
– In fact, for 0 < x < 1 we have √x > x, not the other way around.
⇒ Statement II is false.
Therefore, only statement I is correct.
