Solution:
We start by inferring from the question stem, which states that X > Y^2 > Z^4. This implies:
• X must be greater than Y^2, and Y^2 must be greater than Z^4, X > 0 , Y^2 > 0
• Since squares and fourth powers of numbers between 0 and 1 are smaller than the original numbers, we should consider values for Y and Z that are between 0 and 1 for some cases to verify each statement.
Checking Each Statement:
1. Statement I: X > Y > Z
values: X = 2, Y = 1, Z = 0
Check X > Y^2 > Z^4:
so 2 > 1 > 0, which satisfies X > Y^2 > Z^4.
Conclusion: Statement I could be true.
2. Statement II: Z > Y > X
Values: X = 1/4, Y = 1/3, Z = 1/2
Check X > Y^2 > Z^4:
Y^2 = (1/3)^2 = 1/9 and Z^4 = (1/2)^4 = 1/16, so 1/4 > 1/9 > 1/16, which satisfies X > Y^2 > Z^4.
Check Z > Y > X:
1/2 > 1/3 > 1/4, which holds.
Conclusion: Statement II could be true.
3. Statement III: X > Z > Y
Values: X = 1, Z = 1/3, Y = 1/4
Check X > Y^2 > Z^4:
Y^2 = (1/4)^2 = 1/16 and Z^4 = (1/3)^4 = 1/81, so 1 > 1/16 > 1/81, which satisfies X > Y^2 > Z^4.
Check X > Z > Y:
1 > 1/3 > 1/4, which holds.
Conclusion: Statement III could be true.
Final Conclusion: Answer: E