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the correct ordering

by sanju09 » Fri Mar 23, 2012 3:25 am
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
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by killer1387 » Fri Mar 23, 2012 3:59 am
sanju09 wrote: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.
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by Anurag@Gurome » Fri Mar 23, 2012 4:20 am
sanju09 wrote: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.
Last edited by Anurag@Gurome on Fri Mar 23, 2012 4:29 am, edited 1 time in total.
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by killer1387 » Fri Mar 23, 2012 4:23 am
Anurag@Gurome wrote:
sanju09 wrote: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.
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by Anurag@Gurome » Fri Mar 23, 2012 4:29 am
killer1387 wrote: @ 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.
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