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Is A positive? 1. x^2 - 2x + A is +ve for all x 2. (A*x^2) +

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Is A positive? 1. x^2 - 2x + A is +ve for all x 2. (A*x^2) + [#permalink] New post 21 Mar 2011, 21:52
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

15% (03:29) correct 85% (01:41) wrong based on 13 sessions
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x
[Reveal] Spoiler: OA
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Re: algebra problem [#permalink] New post 21 Mar 2011, 22:23
asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x



x^2 - 2x + A >0
x^2-2x +1 +A-1>0
(X-1)^2 -1 >-A
A>1-(X-1)^2 (flip the sign)
if x is 1/2 A is positive no ...
if x is 5 , A>-15 so A could be either positive or negative

2
A x^2 +1>0
Ax^2>-1
A=-1/2 ,x=1 -1/2>-1 true
A=1 X=2 4>-1 true..
A could be negative eg (-1/2) or positive ... Insufficient..

1+2, Insufficient
E

Am I missing sth
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Re: algebra problem [#permalink] New post 22 Mar 2011, 00:45
i think the answer should be E instead of A
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Re: algebra problem [#permalink] New post 22 Mar 2011, 01:21
I picked numbers and got E, basically the goal is to prove that the expression is +ve for both -ve and +ve values for A.

(1) is not sufficient

3^2 - 2*3 + 1/2 = +ve

3^2 - 2*3 - 1/2 = +ve

A can be +ve or -ve

(2)

Ax^2 is +ve and 1 is +ve

-1/2 * (1/2)^2 + 1 is +ve

1/2 * (-1/2)^2 + 1 is +ve

So (2) is not sufficient

From(1) and (2)

(-1/2)^2 - 2 * -1/2 - 1/2 is +ve

(-1/2)^2 - 2 * -1/2 + 1/2 is +ve

Answer ie E.
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Re: algebra problem [#permalink] New post 22 Mar 2011, 01:58
asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x


We need to answer the question "Is A positive?"

Statement 1 says x^2 - 2x + A is positive for all x. Hence A can only take values that make the expression x^2 - 2x + A positive for all x.

Now, the term x^2-2x will be always positive for all x greater than 2. Also, it will always be positive for all x less than zero and it will be zero in case x=0. So for all x greater than or equal to 2 and less than or equal to zero, A can be negative or positive and still the expression can be positive overall depending on magnitude of a and A. So, this does not give us a condition on A.

Lets see when x lies between 0 and 2. In such a case, for both x =0 and x=2, the expression x^2-2x is zero and hence A needs to be positive. Between x=0 and x=2, the expression remains negative with minimum value of -1 at x=1. Hence, A needs to be greater than +1 to ensure that x^2-2x+A is greater than zero for all x when x lies between 0 and 2.

So, we have seen that for all x, the expression x^2 - 2x + A is positive if and only if A is greater than 1 always. Hence we can answer the question "Is A positive?" in affirmative and hence sufficient.

Statement 2 says that A*x^2+1 is always positive for all x. Clearly, x^2 is always positive. So, if A is always positive than A*x^2+1 is always positive. However, A*x^2+1is always positive also when A=0. Hence, Statement 2) will hold when A is 0 or positive and hence we cannot answer for sure the question "Is A positive?". Hence, insufficient.

Answer is A.
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Re: algebra problem [#permalink] New post 22 Mar 2011, 18:30
1. sufficient, as A can only be +ve to satisfy the given condition.
2. Not sufficient as we could have A>0 or A=0


Answer is A.
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Re: algebra problem [#permalink] New post 22 Mar 2011, 18:40
I agree with beyondgmatscore. A has to be positive to satisfy the expression for all values of x.

beyondgmatscore wrote:
asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x


We need to answer the question "Is A positive?"

Statement 1 says x^2 - 2x + A is positive for all x. Hence A can only take values that make the expression x^2 - 2x + A positive for all x.

Now, the term x^2-2x will be always positive for all x greater than 2. Also, it will always be positive for all x less than zero and it will be zero in case x=0. So for all x greater than or equal to 2 and less than or equal to zero, A can be negative or positive and still the expression can be positive overall depending on magnitude of a and A. So, this does not give us a condition on A.

Lets see when x lies between 0 and 2. In such a case, for both x =0 and x=2, the expression x^2-2x is zero and hence A needs to be positive. Between x=0 and x=2, the expression remains negative with minimum value of -1 at x=1. Hence, A needs to be greater than +1 to ensure that x^2-2x+A is greater than zero for all x when x lies between 0 and 2.

So, we have seen that for all x, the expression x^2 - 2x + A is positive if and only if A is greater than 1 always. Hence we can answer the question "Is A positive?" in affirmative and hence sufficient.

Statement 2 says that A*x^2+1 is always positive for all x. Clearly, x^2 is always positive. So, if A is always positive than A*x^2+1 is always positive. However, A*x^2+1is always positive also when A=0. Hence, Statement 2) will hold when A is 0 or positive and hence we cannot answer for sure the question "Is A positive?". Hence, insufficient.

Answer is A.
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Re: algebra problem [#permalink] New post 22 Mar 2011, 19:30
beyondgmatscore wrote:
asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x


We need to answer the question "Is A positive?"

Statement 1 says x^2 - 2x + A is positive for all x. Hence A can only take values that make the expression x^2 - 2x + A positive for all x.

Now, the term x^2-2x will be always positive for all x greater than 2. Also, it will always be positive for all x less than zero and it will be zero in case x=0. So for all x greater than or equal to 2 and less than or equal to zero, A can be negative or positive and still the expression can be positive overall depending on magnitude of a and A. So, this does not give us a condition on A.

Lets see when x lies between 0 and 2. In such a case, for both x =0 and x=2, the expression x^2-2x is zero and hence A needs to be positive. Between x=0 and x=2, the expression remains negative with minimum value of -1 at x=1. Hence, A needs to be greater than +1 to ensure that x^2-2x+A is greater than zero for all x when x lies between 0 and 2.

So, we have seen that for all x, the expression x^2 - 2x + A is positive if and only if A is greater than 1 always. Hence we can answer the question "Is A positive?" in affirmative and hence sufficient.

Statement 2 says that A*x^2+1 is always positive for all x. Clearly, x^2 is always positive. So, if A is always positive than A*x^2+1 is always positive. However, A*x^2+1is always positive also when A=0. Hence, Statement 2) will hold when A is 0 or positive and hence we cannot answer for sure the question "Is A positive?". Hence, insufficient.

Answer is A.

Assume x =5
x^2 - 2x=> 25-10 =15
for x^2 - 2x + A to be positive A can be either positive no or negative no
if A= -1 (Statement A is still valid x^2 - 2x + A>0)
if A=1 (Statement A is still valid x^2 - 2x + A>0)
A can be either positive no or negative no ..So How could you say A>0 ....Am I missing something ?
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Re: algebra problem [#permalink] New post 22 Mar 2011, 19:42
I think the designer would have rephrased the question as- if I make x=0 then will the "A" stand the test of being positive.

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Re: algebra problem [#permalink] New post 22 Mar 2011, 21:34
Onell wrote:
beyondgmatscore wrote:
asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x


We need to answer the question "Is A positive?"

Statement 1 says x^2 - 2x + A is positive for all x. Hence A can only take values that make the expression x^2 - 2x + A positive for all x.

Now, the term x^2-2x will be always positive for all x greater than 2. Also, it will always be positive for all x less than zero and it will be zero in case x=0. So for all x greater than or equal to 2 and less than or equal to zero, A can be negative or positive and still the expression can be positive overall depending on magnitude of a and A. So, this does not give us a condition on A.

Lets see when x lies between 0 and 2. In such a case, for both x =0 and x=2, the expression x^2-2x is zero and hence A needs to be positive. Between x=0 and x=2, the expression remains negative with minimum value of -1 at x=1. Hence, A needs to be greater than +1 to ensure that x^2-2x+A is greater than zero for all x when x lies between 0 and 2.

So, we have seen that for all x, the expression x^2 - 2x + A is positive if and only if A is greater than 1 always. Hence we can answer the question "Is A positive?" in affirmative and hence sufficient.

Statement 2 says that A*x^2+1 is always positive for all x. Clearly, x^2 is always positive. So, if A is always positive than A*x^2+1 is always positive. However, A*x^2+1is always positive also when A=0. Hence, Statement 2) will hold when A is 0 or positive and hence we cannot answer for sure the question "Is A positive?". Hence, insufficient.

Answer is A.

Assume x =5
x^2 - 2x=> 25-10 =15
for x^2 - 2x + A to be positive A can be either positive no or negative no
if A= -1 (Statement A is still valid x^2 - 2x + A>0)
if A=1 (Statement A is still valid x^2 - 2x + A>0)
A can be either positive no or negative no ..So How could you say A>0 ....Am I missing something ?


Onell - you are right in deducing that for x=5 A can be either positive or negative. In fact, if you see my response, I have reached a similar conclusion as well. However, we need the value of A for which the expression x^2 - 2x + Ais positive for ALL x. As I explained earlier, for x between 0 and 2, this can be true only if A is greater than 1. So, x^2 - 2x + A is positive for ALL x only when A is greater than 1. For some x, it can be positive even when A is negative, but it is positive for ALL x if and only if A is greater than +1.

Does this clarify your doubt now?
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Re: algebra problem [#permalink] New post 07 Apr 2011, 08:57
The first statement can be solved with some calculus. Differentiating x^2 - 2x + A yields 2x - 2. If we assume 2x - 2 = 0 and x = 1, we know that the quadratic curve is at its minimum (lowest value possible) when x = 1.

Therefore, substituting x = 1 into x^2 - 2x + A, we get -1 + A. This is the lowest value possible and since -1 + A > 0 (always positive), A > 1 and always positive.

So it's sufficient.

I'm inclined to say the second statement is true as well. If A < 0, the quadratic curve will have a maximum, which means it will definitely go negative, hence contradicting the statement ("(A*x^2) + 1 is +ve for all x "). The only way to make it always positive is if A > 0.

So I personally think the answer should be D.

I don't have any software to plot graphs right now, but if anyone fancies plotting quadratic graphs they might be able to show this.
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Re: algebra problem [#permalink] New post 07 Apr 2011, 21:54
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The first statement can be solved with some calculus. Differentiating x^2 - 2x + A yields 2x - 2. If we assume 2x - 2 = 0 and x = 1, we know that the quadratic curve is at its minimum (lowest value possible) when x = 1.

Therefore, substituting x = 1 into x^2 - 2x + A, we get -1 + A. This is the lowest value possible and since -1 + A > 0 (always positive), A > 1 and always positive.

So it's sufficient.

I'm inclined to say the second statement is true as well. If A < 0, the quadratic curve will have a maximum, which means it will definitely go negative, hence contradicting the statement ("(A*x^2) + 1 is +ve for all x "). The only way to make it always positive is if A > 0.

So I personally think the answer should be D.

I don't have any software to plot graphs right now, but if anyone fancies plotting quadratic graphs they might be able to show this.


Your reasoning for the second statement is incorrect. The only way to make it always positive is if A > 0 OR A = 0. As, at A=0, the value of expression is 1 - always positive. So, A can be zero or positive and hence insufficient.
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Re: algebra problem [#permalink] New post 08 Apr 2011, 01:52
beyondgmatscore wrote:

Your reasoning for the second statement is incorrect. The only way to make it always positive is if A > 0 OR A = 0. As, at A=0, the value of expression is 1 - always positive. So, A can be zero or positive and hence insufficient.


Ah yes, I overlooked that. Thanks!
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Re: algebra problem [#permalink] New post 08 Apr 2011, 06:28
asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x


I've posted about this question a couple of times. Many people solve this backwards and arrive at the wrong answer (those test takers relying on 'number picking' strategies almost always answer this question incorrectly). The important word in each of the two statements is 'all'. In Statement 1, x^2 - 2x + A is positive not just for some value of x; it must be positive for EVERY value of x. In particular, it's positive when x=0, so substituting x=0, we learn instantly that A is positive and Statement 1 is sufficient.

Statement 2 is also almost sufficient. It is only insufficient because of a technicality: it's possible that A=0.

So the answer is A.
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Re: algebra problem   [#permalink] 08 Apr 2011, 06:28
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