the correct ordering

[sanju09]

If x is a positive integer, then which of the following could be the correct ordering of 1/x, 2 x, and x^2?
I. x^2 < 2 x < 1/x
II. x^2 < 1/x < 2 x
III. 2 x < x^2 < 1/x

A. None
B. III only
C. I only
D. I and II only
E. I, II, and III

[killer1387]

If x is a positive integer, then which of the following could be the correct ordering of 1/x, 2 x, and x^2?
I. x^2 < 2 x < 1/x
II. x^2 < 1/x < 2 x
III. 2 x < x^2 < 1/x

A. None
B. III only
C. I only
D. I and II only
E. I, II, and III

IMO A.

[Anurag@Gurome]

If x is a positive integer, then which of the following could be the correct ordering of 1/x, 2 x, and x^2?
I. x^2 < 2 x < 1/x
II. x^2 < 1/x < 2 x
III. 2 x < x^2 < 1/x

A. None
B. III only
C. I only
D. I and II only
E. I, II, and III

I. x² < 2x < 1/x
If x = 2, then x² = 4, 2x = 2 and 1/x = 1/2 implies 4 < 2 < 1/2, which is NOT the correct ordering.

II. x² < 1/x < 2x
If x = 3, then x² = 9, 2x = 6 and 1/x = 1/3 implies 9 < 1/3 < 6, which is NOT the correct ordering again.

III. 2x < x² < 1/x
2x < x² implies 2x - x² < 0 or x(2 - x) < 0 implies x > 2
If x > 2, then 1/x is the least value of all the three, which is not so. So, III cannot be true.

The correct answer is A.

[killer1387]

If x is a positive integer, then which of the following could be the correct ordering of 1/x, 2 x, and x^2?
I. x^2 < 2 x < 1/x
II. x^2 < 1/x < 2 x
III. 2 x < x^2 < 1/x

A. None
B. III only
C. I only
D. I and II only
E. I, II, and III

I. x² < 2x < 1/x
If x = 1/2, then x² = 1/4, 2x = 1 and 1/x = 2 implies 1/4 < 1 < 2, which is the correct ordering.

II. x² < 1/x < 2x
If x = 4/5, then x² = 16/25, 2x = 8/5 and 1/x = 5/4 implies 16/25 < 5/4 < 8/5, which is the correct ordering.

III. 2x < x² < 1/x
2x < x² implies 2x - x² < 0 or x(2 - x) < 0 implies x > 2
If x > 2, then 1/x is the least value of all the three, which is not so. So, III cannot be true.

The correct answer is D.

@ Anurag

the question stem mentions X is a positive integer.
Hence in no case will 1/x i.e. a fraction be greater than 2x or x^2, i.e. an integer.

[Anurag@Gurome]

@ Anurag

the question stem mentions X is a positive integer.
Hence in no case will 1/x i.e. a fraction be greater than 2x or x^2, i.e. an integer.

Ok, I missed, I took x as any positive number. I've edited my previous reply.