VeritasPrepKarishma wrote:

rongali wrote:

Is A positive?

1) X^2-2X+A is positive for all X

2) AX^2 + 1 is positive for all X

given answer as A...but i thought it should be E..

source: hard problems from

gmatclub tests number properties I

1) X^2-2X+A is positive for all X

For all values of X,\(X^2-2X+A > 0\)

This means, for X = 0, \(X^2-2X+A > 0\); for X = 1, \(X^2-2X+A > 0\); for X = -2, \(X^2-2X+A > 0\) etc etc etc

Let's put X = 0. \(0^2-2*0+A > 0\) should hold. Therefore, A > 0 should hold.

Sufficient.

2) AX^2 + 1 is positive for all X

For all X, \(AX^2 + 1 > 0\)

Here, A could be positive or A could be 0 (since, when A = 0, we get 1 > 0 which holds no matter what the value of X.)

Since A can be 0, we cannot say whether A is positive. Not Sufficient.

Answer A

Responding to a pm:

**Quote:**

I still did not understand your solution

x^2-2x+A>0

if we take the value 3 for example ,

9-6+A>0

3+A>0

which gives

A>(-)3

so A can assume -2,-1,0 and so on and we still get the overall value as +ve.

Can you help me understand what i am missing ?

Given that x^2-2x+A is always positive. No matter what the value of x, the value of A is such that this expression is always positive.

Whether x = ...-2, 0, 1, 4, 100..., the expression will always be positive.

So let's put a few values of x.

Put x = -2

(-2)^2-2(-2)+A > 0

A > -8

Put x = 0

0^2 - 2*0 + A > 0

A > 0

Put x = 1

1^2 - 2*1 + A > 0

A > 1

Put x = 3

3^2 - 2*3 + A > 0

A > -3

and so on...

So we see that A must be greater than -8, it should also be greater than -3, it should also be greater than 0 and it should also be greater than 1. So what values do you think A can take? Values which are greater than all these values i.e. values like 8, 10 etc. In any case, we are asked whether A is positive and we know that it must be greater than 1. Hence, we know that A must be positive. Sufficient.

_________________

Karishma

Veritas Prep | GMAT Instructor

My Blog

Get started with Veritas Prep GMAT On Demand for $199

Veritas Prep Reviews