is A positive?

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gmatrix: Master | Next Rank: 500 Posts; Posts: 108; Joined: Fri Jul 09, 2010 8:15 am; Location: 127.0.0.1; Thanked: 15 times

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: Beat The GMAT — Data Sufficiency | Fri Jul 30, 2010 8:17 pm

selango: Legendary Member; Posts: 1460; Joined: Tue Dec 29, 2009 1:28 amFollowed by: 7 members; Thanked: 135 times

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

--Anand--

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

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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gmatrix: Master | Next Rank: 500 Posts; Posts: 108; Joined: Fri Jul 09, 2010 8:15 am; Location: 127.0.0.1; Thanked: 15 times

by gmatrix » Fri Jul 30, 2010 10:11 pm

source:gmat club tests
OA: A

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

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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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

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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gmat1011: Master | Next Rank: 500 Posts; Posts: 160; Joined: Mon Apr 05, 2010 4:41 am; Thanked: 7 times

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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gmatrix: Master | Next Rank: 500 Posts; Posts: 108; Joined: Fri Jul 09, 2010 8:15 am; Location: 127.0.0.1; Thanked: 15 times

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

)...
https://gmatclub.com/forum/is-a-positive ... ml#p755641

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