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is A positive?

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by gmatrix » Fri Jul 30, 2010 8:17 pm
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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Source: — Data Sufficiency |
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by selango » Fri Jul 30, 2010 8:50 pm
stmt1,

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

x=1,1-2+A=-1+A,A is +ve

x=-1,1+2+A=3+A.A can be zero,+ve or -ve

Insuff

stmt2,

Ax^2+1 is positive

-->Ax^2>=0

x^2 is always +ve

A>=0

A can be o or +ve.

Insuff

Combining `1 and 2,

Still A>=0

A can be o or +ve.

Insuff

Hence E
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by Stuart@KaplanGMAT » Fri Jul 30, 2010 8:57 pm
gmatrix wrote:is A positive?

1.x^2-2x+A is positive for all x
2.Ax^2+1 is positive for all x
Tricky question! What's the source? (You should post the source for every question!)

Yes/no question, so if A is always positive, sufficient; if A is never positive, sufficient; if sometimes A is positive and sometimes not, insufficient.

Let's start with (2), since that's simpler:

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

Well, x^2 is always greater than or equal to 0. So, Ax^2 can only be negative if A is negative.

So, for Ax^2 + 1 to always be positive:

Ax^2 > -1

A > -1/x^2


If x^2 = 4, then x = 2 and A > -1/4; so, A could be negative OR positive: insufficient.


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

So:

x^2 - 2x + A > 0

A > 2x - x^2

A > 2(1 - x)

if x = 2, then A > -2 (could be positive or negative)

Insufficient!

Together:

if x = 2, A could be negative OR positive and satisfy both statements; accordingly, insufficient. Choose (E).
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by gmatrix » Fri Jul 30, 2010 10:11 pm
source:gmat club tests
OA: A
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by Ian Stewart » Fri Jul 30, 2010 10:12 pm
gmatrix wrote:is A positive?

1.x^2-2x+A is positive for all x
2.Ax^2+1 is positive for all x
I think the solutions above misinterpret the question. If we know from Statement 1 that x^2 - 2x + A is positive for *all* x, then it certainly is positive when x=0. Substitute x=0 into the inequality and we instantly find that A > 0, so Statement 1 is sufficient.

Statement 2 is not sufficient, but only on a technicality: A can be zero, as you can see by substituting A=0 into the inequality.

I'd add that if Statement 2 is true, A cannot be negative. The easiest way for me to see that is through coordinate geometry, but this goes beyond the scope of the GMAT: if A is nonzero, the equation y = Ax^2 + 1 represents a parabola (a U shape, pointing either upwards or downwards) with a y-intercept at (0,1). If A is negative, this parabola will slope downwards, so as you move to the right or left of x=0, it will eventually fall below the x-axis, so y will be negative for some (well, infinitely many, in fact) values of x. So if Ax^2 + 1 is *always* positive for any value of x, A must be zero or greater.
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by Stuart@KaplanGMAT » Fri Jul 30, 2010 10:45 pm
Ian Stewart wrote:
gmatrix wrote:is A positive?

1.x^2-2x+A is positive for all x
2.Ax^2+1 is positive for all x
I think the solutions above misinterpret the question. If we know from Statement 1 that x^2 - 2x + A is positive for *all* x, then it certainly is positive when x=0. Substitute x=0 into the inequality and we instantly find that A > 0, so Statement 1 is sufficient.

Statement 2 is not sufficient, but only on a technicality: A can be zero, as you can see by substituting A=0 into the inequality.

I'd add that if Statement 2 is true, A cannot be negative. The easiest way for me to see that is through coordinate geometry, but this goes beyond the scope of the GMAT: if A is nonzero, the equation y = Ax^2 + 1 represents a parabola (a U shape, pointing either upwards or downwards) with a y-intercept at (0,1). If A is negative, this parabola will slope downwards, so as you move to the right or left of x=0, it will eventually fall below the x-axis, so y will be negative for some (well, infinitely many, in fact) values of x. So if Ax^2 + 1 is *always* positive for any value of x, A must be zero or greater.
On reflection, you're right about the misinterpretation!
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by gmat1011 » Fri Jul 30, 2010 11:00 pm
I remember from one of Stuart's earlier posts (all very helpful:)

that if we see a quad eq of the form x^2-2x+A then looking at the sign of the constant (A in this case) one can decipher whether the roots will carry the same sign or carry opposite signs.... If X is always positive then A has to carry a +sign (if its -ve roots will be -ve and +ve, i.e., with opposite signs).

Based on that can one conclude its A?
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by gmatrix » Sat Jul 31, 2010 1:25 am
I think the solutions above misinterpret the question.
hmmm.......the explaination within the test was sketchy to say the least...but it's solution is explained in detail here(I should learn to search :oops:)...
https://gmatclub.com/forum/is-a-positive ... ml#p755641
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