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# Is A positive? 1) x^2 - 2x +A is positive for all x 2) A*x^2

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Director
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Is A positive? 1) x^2 - 2x +A is positive for all x 2) A*x^2 [#permalink]  17 Sep 2007, 22:55
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Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.
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St1:
x^2-2x+A is positive for all x.

If x=1/2, then x^2-2x+A = -3/4 + A. So A -> positive if x^2-2x+A is positive.
If x=5, then x^2-2x+A = 15+A. So A -> positive or A is negative but bigger than -15. However, since we're saying that it must be all x, A must be positive otherwise fractional x wouldn't work. Sufficient.

St2:
A*x^2 + 1 is positive for all x

If x=1/2, then A*x^2 + 1 = A/4 + 1. A can be negative or positive and A*x^2 + 1 will be positive.

If x=2, then A*x^2 + 1 = 4a + 1. A has to be positive if A*x^2 + 1 is positive. Since we want A*x^2 + 1 to be positive for all x, then A has to be positive if not integer values of x won't work. Sufficient.

Ans D
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ywilfred wrote:
St1:
x^2-2x+A is positive for all x.

If x=1/2, then x^2-2x+A = -3/4 + A. So A -> positive if x^2-2x+A is positive.
If x=5, then x^2-2x+A = 15+A. So A -> positive or A is negative but bigger than -15. However, since we're saying that it must be all x, A must be positive otherwise fractional x wouldn't work. Sufficient.

St2:
A*x^2 + 1 is positive for all x

If x=1/2, then A*x^2 + 1 = A/4 + 1. A can be negative or positive and A*x^2 + 1 will be positive.

If x=2, then A*x^2 + 1 = 4a + 1. A has to be positive if A*x^2 + 1 is positive. Since we want A*x^2 + 1 to be positive for all x, then A has to be positive if not integer values of x won't work. Sufficient.

Ans D

I disagree with D.

From stmt 1 we know A > 2x. This doesnt mean that A is positive or negative.
From Stmt 2 we know that A > -1/x^2. So no clue here too

Comparing the statements I feel it is E.
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 12:55
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

i think it is E

1) A could be positive or negative
ex. x = 4, A = -5 -> 4^2 - 2(4) - 5 = 3
stmt still true

ex. x = 4, A = 5 -> 4^2 - 2(4) + 5 = 13
stmt still true

2) A could be either again since does not specify whether int or not
x^2 is always positive but A can be neg (if the product for A* x^2 is -1/2 for instance.. stmt still holds true) or A can positive and yield a positive answer
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 13:30
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

ST1: x^2 - 2x +A is positive for all x

Let's say x = -1

Then, 1 + 2 + A > 0
=> 3 + A > 0
=> A > (-3)

So A may be -2 , -1 , 0 or > 0

Hence A may be positive or may be negative.

NOT SUFF.

ST2: A*x^2 + 1 is positive for all x

Again let's say x = -1

Then, A + 1 >0
=> A > (-1)

So A may be 0 which is neither positive nor negative.

NOT SUFF.

Taking ST1 and ST2 together, I am still not able to arrive at any conclusion.

So in my opinion, answer should be E.

- Brajesh
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similar to others this is how I got E:

a) x^2 - 2x+A

let X =4

16 - 8 + A > 0, "A" could be 2. "A" could also be -6. Not Sufficient.

b) A*X^2 + 1

X^2 is always positive. Let X = 4:

16A + 1. "A" could be 2. "A" could also be Zero. Not sufficient.

combined however, I just don't see any correlation between the two. So I'd go with E as well.
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 14:29
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

1: x^2 - 2x +A > 0

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 14:51
Fistail wrote:
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

1: x^2 - 2x +A > 0

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.

Indeed a good approach Fistail.

But how about putting the value of x = -1 and obtaining the possible values of A as I mentioned in my above earlier post?

Writing again:

ST1: x^2 - 2x +A is positive for all x

Let's say x = -1

Then, 1 + 2 + A > 0
=> 3 + A > 0
=> A > (-3)

So A may be -2 , -1 , 0 or > 0

Hence A may be positive or negative.

- Brajesh
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 15:05
b14kumar wrote:
Fistail wrote:
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

1: x^2 - 2x +A > 0

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.

Indeed a good approach Fistail.

But how about putting the value of x = -1 and obtaining the possible values of A as I mentioned in my above earlier post?

Writing again:

ST1: x^2 - 2x +A is positive for all x

Let's say x = -1

Then, 1 + 2 + A > 0
=> 3 + A > 0
=> A > (-3)

So A may be -2 , -1 , 0 or > 0

Hence A may be positive or negative.

- Brajesh

I again looked at your approach.

As per you:

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0.
if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.
Why are you assuming only the case when (x-1)^2 is equal to 0?

Well, (x -1)^2 will always be >= 0 but it does not mean that "A – 1" has to be positive for all the cases.
Imagine, (x -1)^2 is equal to 4 (by taking x = 3 ) , in this case, (A-1) can be (-3) i.e A can be (-2) and still the whole inequality will be intact.

- Brajesh
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 15:27
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.
I get D.

x^2 - 2x + A
the bolded portion will always be even. in order for this whole expression to be even won't A also have to be even?

A*x^2 + 1
the bolded portion will always be even. in order for the entire expression to be +ve A will also have to be +ve.

what is the OA and OE?
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 16:19
ggarr wrote:
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.
I get D.

x^2 - 2x + A
the bolded portion will always be even. in order for this whole expression to be even won't A also have to be even?

I think you misread the question. It asks whether A is POSITIVE not EVEN.
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 19:52
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

After a many computations I arrived at E. Since A could be a fraction

S1 and S2 are insuff. Lookin at the dispute over this prob I will put down wut I have from my notes.

S1: x=3, A=2 3^2-6+2 =+
x=3 A=-2 3^2-6-2=1 so its still + but A is -. Insuff.

S2: x=-1/2 a=-1/4 -1/4*-1/2^2---> -1/4*1/4---> -1/16+1 is positive.

x=3 a =2 2*3^2+1 = positive. So insuff.

S1&S2:
S1:x=-1/2 A=-1/4 -1/2^2-2(-1/2) +(-1/4) ---> 1/4+1 -1/4 = positive.

A is -.
S2: x=-1/2 a=-1/4 -1/4*-1/2^2---> -1/4*1/4---> -1/16+1 is positive.

A is -.

S1: x=3 a =2. Again same process as above.

So A can be + or - in both situations.

ANS E.
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Re: DS: Is A positive? [#permalink]  18 Sep 2007, 22:00
b14kumar wrote:
b14kumar wrote:
Fistail wrote:
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

1: x^2 - 2x +A > 0

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.

Indeed a good approach Fistail.

But how about putting the value of x = -1 and obtaining the possible values of A as I mentioned in my above earlier post?

Writing again:

ST1: x^2 - 2x +A is positive for all x

Let's say x = -1

Then, 1 + 2 + A > 0
=> 3 + A > 0
=> A > (-3)

So A may be -2 , -1 , 0 or > 0

Hence A may be positive or negative.

- Brajesh

I again looked at your approach.

As per you:

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0.
if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.
Why are you assuming only the case when (x-1)^2 is equal to 0?

Well, (x -1)^2 will always be >= 0 but it does not mean that "A – 1" has to be positive for all the cases.
Imagine, (x -1)^2 is equal to 4 (by taking x = 3 ) , in this case, (A-1) can be (-3) i.e A can be (-2) and still the whole inequality will be intact.

- Brajesh

The OA is A. But it doesn't make sense. I'm having the same problem understanding the solution as Brijesh (Above). Any thoughts?
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Re: DS: Is A positive? [#permalink]  19 Sep 2007, 01:40
GK_Gmat wrote:
Is A positive?

1) x^2 - 2x +A is positive for all x
2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

My explanation:

from (i)

x^2-2*x+A > 0

let's assume X as +

then A has to be positive for above inequality to be true.

Let's assume X as -

then also A has to be positive for above inequality to be true.

No worries it's fraction or not. ( postive or negative is the key )

from ( ii )

A*x^2+1>0

which implies A > -1/x^2

depending on the value of x which might be ( + or - ) or fraction, A might be..anywhere on the number line.

Therefore the answer should be A.

Any questions regarding the explanation are welcome.
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The catch in this question is that A is CONSTATNT not a variable. So for all x, A will have the same value.

Now read it as - Find the value of A (+/-) for which x^2 - 2x + A > 0 is true for all x.
Paraphrase this (x-1)^2 + A-1 > 0
The minimum value of (x-1)^2 is zero. So A > 1.
So A> 1 satisfies the expression for all x.
Hence SUFF.

Stmt2: Ax^2 + 1 > 0
If we take A -ve, the above expression will be true for some cases and false for some cases depending on value of x. To make this true for all x, A should be +ve or zero.
So INSUFF.

Last edited by vshaunak@gmail.com on 21 Sep 2007, 03:02, edited 1 time in total.
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vshaunak@gmail.com wrote:
The catch in this question is that A is CONSTATNT not a variable. So for all x, A will have the same value.

Now read it as - Find the value of A (+/-) for which x^2 - 2x + A > 0 is true for all x.
Paraphrase this (x-1)^2 + A-1 > 0
The minimum value of (x-1)^2 is zero. So A > 1.
So A> 1 satisfies the expression for all x.
Hence SUFF.

Stmt2: Ax^2 + 1 > 0
If we take A -ve, the above expression will be true for some cases and false for some cases depending on value of x. To make this true for all x, A should be +ve.
So SUFF.

I think OA is correct as A .
For Stmtn 1 - same as marked as blue .

For Stmtn 2 - you can never determine whether A is +ve or -ve

Ax^2 + 1 = 0.5 => A = -ve
Ax^2 +1 = 2 => A = +ve
So stmtn is not suff .
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hmmmm how would we know whether A is constant? now that you've pointed it out, I guess I see it, but it would be very tough to catch on the real GMAT>
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Out of all, Brijesh's explanation makes sense to me where he says A could be even -1 or -2 and still be bigger than -3.
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I am new here:

OE = Official Explanation???
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I get E too...

i think if you assume A to be a constant then statement 2 would be sufficient as well..then .the OA should have been D..

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