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Is A positive? 1) x^2  2x +A is positive for all x 2) A*x^2 [#permalink]
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17 Sep 2007, 23:55
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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^22x+A is positive for all x.
If x=1/2, then x^22x+A = 3/4 + A. So A > positive if x^22x+A is positive.
If x=5, then x^22x+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^22x+A is positive for all x.
If x=1/2, then x^22x+A = 3/4 + A. So A > positive if x^22x+A is positive. If x=5, then x^22x+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]
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18 Sep 2007, 13:55
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]
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18 Sep 2007, 14:30
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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Joined: 15 Sep 2007
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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]
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18 Sep 2007, 15:29
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]
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18 Sep 2007, 15:51
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]
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18 Sep 2007, 16:05
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 (x1)^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, (A1) can be (3) i.e A can be (2) and still the whole inequality will be intact.
Please let me know your opinion.
 Brajesh



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Re: DS: Is A positive? [#permalink]
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18 Sep 2007, 16:27
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]
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18 Sep 2007, 17:19
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]
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18 Sep 2007, 20:52
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 3^26+2 =+
x=3 A=2 3^262=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 1/2^22(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]
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18 Sep 2007, 23:00
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 (x1)^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, (A1) can be (3) i.e A can be (2) and still the whole inequality will be intact. Please let me know your opinion.  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]
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19 Sep 2007, 02:40
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^22*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.



Director
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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 (x1)^2 + A1 > 0
The minimum value of (x1)^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.
My answer is 'A'



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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 (x1)^2 + A1 > 0 The minimum value of (x1)^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.
My answer is 'D'
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.
How about posting OE?



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I am new here:
OA = Official Answer ???
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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