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# Is A positive? 1. x^2 - 2x + A is positive for all x 2. Ax^2

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Is A positive? 1. x^2 - 2x + A is positive for all x 2. Ax^2 [#permalink]

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01 Sep 2008, 04:31
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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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Re: A+VE [#permalink]

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01 Sep 2008, 07:32
I'd go for A.

S1) A = +1. SUFF.
S2) A(x^2)+1 = POS ==> A(x^2) = POS-1, thus A(x^2) could equal a positive number or zero. A could be any positive number or zero, which is not positive. INSUFF.

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Re: A+VE [#permalink]

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01 Sep 2008, 10:48
how do u say 1 is suff??

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Re: A+VE [#permalink]

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01 Sep 2008, 11:41
asdert wrote:
I'd go for A.

S1) A = +1. SUFF.
S2) A(x^2)+1 = POS ==> A(x^2) = POS-1, thus A(x^2) could equal a positive number or zero. A could be any positive number or zero, which is not positive. INSUFF.

lets say X=5, and A=10
then 1) 25-10+10>0
but if x=5, and A=-10
then 25-10-10>10
same is true for A=0
so A is Insufficient

2) A can be either 0, or A>0, also insufficient
IMO both are also insufficient.

what is the OA?

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Re: A+VE [#permalink]

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01 Sep 2008, 12:02
I solved by undisguising the quadratic, but I may be wrong:
$$(x-1)(x-1) = x^2-2x+1,$$ thus A = 1 = SUFF.

What's the answer?

I hate picking random numbers in problems like this because they don't always work:
$$x^2 - 2x + A$$ is positive for all $$x$$

If A = -10, then result must be positive for all X. So what if X is -2?
$$(-2)^2 - (-4) + (-10) =$$ POSITIVE
$$4 + 4 - 10 =$$NEGATIVE
At least that's what I think.

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Re: A+VE [#permalink]

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01 Sep 2008, 12:33
arjtryarjtry wrote:
Is $$A$$ positive?

1. $$x^2 - 2x + A$$ is positive for all $$x$$
2. $$Ax^2 + 1$$ is positive for all $$x$$

1) x^2 - 2x + A >0

A> -x^2+2x -- true for all values of x.

please note that here A is constant and x is variable..

Find out the possible values of "-x^2+2x"

"-x^2+2x" +ve when X=1 --(1)
"-x^2+2x" -ve when X=3 --(2)

from (1) it is clear that A>+ve so it is positive.
from (2) it is clear that A>-ve so it can positive or negative.

It has to satisfy both equations 1 and 2
A must be positive.

Suffcieint

2)
Ax^2 + 1>0

A> -1/x^2
--> Tells that for all values X A> -ve number.

So A can be +ve or -ve

Insuffcient

A is the Answer.
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Re: A+VE [#permalink]

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01 Sep 2008, 13:04
x2suresh wrote:
arjtryarjtry wrote:
1) x^2 - 2x + A >0

A> -x^2+2x -- true for all values of x.

please note that here A is constant and x is variable..

Find out the possible values of "-x^2+2x"

"-x^2+2x" +ve when X=1 --(1)
"-x^2+2x" -ve when X=3 --(2)

from (1) it is clear that A>+ve so it is positive.
from (2) it is clear that A>-ve so it can positive or negative.

It has to satisfy both equations 1 and 2
A must be positive.

Suffcieint

.

Why does it have to satisfy both equations?

x^2 -2x + A > 0 for all X

x=-1 range of x = { -2, -1 ... infinity} both +ve and -ve

x=-2 range of x = {-7, -6 .. infinity } both -ve and +ve

x= 0 A >0

x=1 A >1

X=2 A>0

X=3 A >-3 again both +ve and -ve

May be is not sufficient

My pick is E

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Re: A+VE [#permalink]

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02 Sep 2008, 04:31
x2suresh wrote:
arjtryarjtry wrote:
Is $$A$$ positive?

1. $$x^2 - 2x + A$$ is positive for all $$x$$
2. $$Ax^2 + 1$$ is positive for all $$x$$

1) x^2 - 2x + A >0

A> -x^2+2x -- true for all values of x.

please note that here A is constant and x is variable..

Find out the possible values of "-x^2+2x"

"-x^2+2x" +ve when X=1 --(1)
"-x^2+2x" -ve when X=3 --(2)

from (1) it is clear that A>+ve so it is positive.
from (2) it is clear that A>-ve so it can positive or negative.

It has to satisfy both equations 1 and 2
A must be positive.

Suffcieint

2)
Ax^2 + 1>0

A> -1/x^2
--> Tells that for all values X A> -ve number.

So A can be +ve or -ve
Insuffcient

A is the Answer.

i dont think A can be negative
because as soon as A becomes the negative, the parabola will be inverted, and you cannot gurantee that for all values of x, $$Ax^2 + 1$$ is positive for all $$x$$[/quote]

but the equation sez other wise.
i am confused !!!

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Re: A+VE [#permalink]

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02 Sep 2008, 05:08
I approached the problem as below.

Stmt 1: x^2 - 2x + A is positive for all x. That means this expression should be positive even for x = 0. With x = 0, the expression changes to A. And, if the expression is positive, A must be positive. Hence, sufficient.

Stmt2: Not sufficient. For a positive value as well as negative value of x, the expression can be positive.

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Re: A+VE [#permalink]

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02 Sep 2008, 06:28
sset009 wrote:
x2suresh wrote:
arjtryarjtry wrote:
Is $$A$$ positive?

1. $$x^2 - 2x + A$$ is positive for all $$x$$
2. $$Ax^2 + 1$$ is positive for all $$x$$

1) x^2 - 2x + A >0

A> -x^2+2x -- true for all values of x.

please note that here A is constant and x is variable..

Find out the possible values of "-x^2+2x"

"-x^2+2x" +ve when X=1 --(1)
"-x^2+2x" -ve when X=3 --(2)

from (1) it is clear that A>+ve so it is positive.
from (2) it is clear that A>-ve so it can positive or negative.

It has to satisfy both equations 1 and 2
A must be positive.

Suffcieint

2)
Ax^2 + 1>0

A> -1/x^2
--> Tells that for all values X A> -ve number.

So A can be +ve or -ve
Insuffcient

A is the Answer.

i dont think A can be negative
because as soon as A becomes the negative, the parabola will be inverted, and you cannot gurantee that for all values of x, $$Ax^2 + 1$$ is positive for all $$x$$

but the equation sez other wise.
i am confused !!! [/quote]

Ax^2 + 1>0

A> -1/x^2

say x=1 A>-1 means A can be -1/2 or zero or Any postive value.
_________________

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Smiling wins more friends than frowning

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Re: A+VE [#permalink]

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02 Sep 2008, 11:46
E for me :

1 is not sufficient ;

x^2-2x + A > 0

here imagine that the value of x is +5 , then the equation becomes 25-10 +A ; now if A is -1/+1 OR -2/+2 etc doesnt make any diff. as the answer will always be
+ve ; same is the case if you take x=-5 .

2 is obv not sufficient as many people have agreed

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Re: A+VE   [#permalink] 02 Sep 2008, 11:46
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# Is A positive? 1. x^2 - 2x + A is positive for all x 2. Ax^2

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