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# Is A positive? 1. x^2 - 2x + A is positive for all x 2. Ax^2

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Joined: 05 Oct 2008
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Is A positive? 1. x^2 - 2x + A is positive for all x 2. Ax^2 [#permalink]

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06 Nov 2008, 05:54
Is $$A$$ positive?

1. $$x^2 - 2x + A$$ is positive for all $$x$$
2. $$Ax^2 + 1$$ is positive for all $$x$$

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SVP
Joined: 29 Aug 2007
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06 Nov 2008, 08:40
study wrote:
Is $$A$$ positive?

1. $$x^2 - 2x + A$$ is positive for all $$x$$
2. $$Ax^2 + 1$$ is positive for all $$x$$

It is one of the most difficult one to grasp. It was discussed just a while ago.

Go though this and then comeback with your doubts: 7-p530279?t=71816#p530279
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SVP
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06 Nov 2008, 22:36
GMAT TIGER wrote:
study wrote:
Is $$A$$ positive?

1. $$x^2 - 2x + A$$ is positive for all $$x$$
2. $$Ax^2 + 1$$ is positive for all $$x$$

It is one of the most difficult one to grasp. It was discussed just a while ago.

Go though this and then comeback with your doubts: 7-p530279?t=71816#p530279

No more takers?

its really intreasting!!!!!!!!!
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VP
Joined: 17 Jun 2008
Posts: 1479

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06 Nov 2008, 23:05
GMAT TIGER wrote:
No more takers?

its really intreasting!!!!!!!!!

Although I had answered E in the previous post, after re-visiting the question, I will go with D. "for all x" is important here.

From stmt1: {m}(x-1)^2 + (A-1) > 0 {/m} for all x and this is possible only when A-1> or A > 1 or A is positive. Sufficient.

From stmt2: A has to be positive as x^2 is positive. Sufficient.
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Joined: 29 Aug 2007
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06 Nov 2008, 23:30
scthakur wrote:
GMAT TIGER wrote:
No more takers?

its really intreasting!!!!!!!!!

Although I had answered E in the previous post, after re-visiting the question, I will go with D. "for all x" is important here.

From stmt1: {m}(x-1)^2 + (A-1) > 0 {/m} for all x and this is possible only when A-1> or A > 1 or A is positive. Sufficient.

From stmt2: A has to be positive as x^2 is positive. Sufficient.

2. Ax^2 + 1 is positive for all x.

What if in statement 2: (1) x = 0.5, A = 1 and (2) x = 0.5, A = -1?
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06 Nov 2008, 23:42
GMAT TIGER wrote:
2. Ax^2 + 1 is positive for all x.

What if in statement 2: (1) x = 0.5, A = 1 and (2) x = 0.5, A = -1?

It is "for all x". Hence, A has to be positive for all x. For x = 0.5, A can be negative , but for x = 50, it cannot be. For all values of x, if A is positive, the inequality will hold good.
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07 Nov 2008, 00:00
scthakur wrote:
GMAT TIGER wrote:
2. Ax^2 + 1 is positive for all x.

What if in statement 2: (1) x = 0.5, A = 1 and (2) x = 0.5, A = -1?

It is "for all x". Hence, A has to be positive for all x. For x = 0.5, A can be negative , but for x = 50, it cannot be. For all values of x, if A is positive, the inequality will hold good.

exactly. here A doesnot need to be +ve to have (Ax^2 + 1) > 0.

With -ve and +ve values for A, (Ax^2 + 1) has to be +ve but not necessarily A.
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07 Nov 2008, 02:15
GMAT TIGER wrote:
exactly. here A doesnot need to be +ve to have (Ax^2 + 1) > 0.

With -ve and +ve values for A, (Ax^2 + 1) has to be +ve but not necessarily A.

I will begin with stmt1.

From stmt1: (x-1)^2 + (A-1) > 0
Here, (x-1)^2 is positive for all X. And, A can be positive or negative for the expression to be true.

For example, x = 0.5, A = 1.
or, x = 5, A = -10
Both the cases will satisfy the ineuqality. However, for certain x, A can be +ve or -ve, but for some other x, A has to be +ve. That means, A must be positive for all x.

From stmt2: Ax^2 + 1 > 0
Here, second expression is positive and x^2 is positive. Hence, A can be positive or negative for certain x and only positive for other x.
For example, for x = 0.5, A can be -1 or +1.
but, for x = 5, A has to be +1 only.
But, positive value of A will satisfy all values of x and not the negative value.
SVP
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07 Nov 2008, 09:19
scthakur wrote:
GMAT TIGER wrote:
exactly. here A doesnot need to be +ve to have (Ax^2 + 1) > 0.

With -ve and +ve values for A, (Ax^2 + 1) has to be +ve but not necessarily A.

I will begin with stmt1.

From stmt1: (x-1)^2 + (A-1) > 0
Here, (x-1)^2 is positive for all X. And, A can be positive or negative for the expression to be true.

For example, x = 0.5, A = 1.
or, x = 5, A = -10
Both the cases will satisfy the ineuqality. However, for certain x, A can be +ve or -ve, but for some other x, A has to be +ve. That means, A must be positive for all x.

From stmt2: Ax^2 + 1 > 0
Here, second expression is positive and x^2 is positive. Hence, A can be positive or negative for certain x and only positive for other x.
For example, for x = 0.5, A can be -1 or +1.
but, for x = 5, A has to be +1 only.
But, positive value of A will satisfy all values of x and not the negative value.

I thik it would be better if we say A can be +ve or 0.

I agree with you that A cannot be -ve.
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07 Nov 2008, 09:59
i agree A is the ans..

stmnt 1: x^2-2x+1+A-1

(x-1)^2 + (A-1)>0 so for all x; if x=0 then its clear A>0
VP
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07 Nov 2008, 11:37
study, do you have the OA and OE (if any) please?

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Re: No. Properties   [#permalink] 07 Nov 2008, 11:37
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# Is A positive? 1. x^2 - 2x + A is positive for all x 2. Ax^2

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