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.
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