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

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

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New post 22 Oct 2013, 21:31
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Raihanuddin wrote:
there are many hard data sufficiency questions? To get over 700 in GMAT at least how many hard data sufficiency questions do we have to answer? I have a lot of problems with hard and tricky DS questions. I always go close to the answer but finally make mistake in hard DS by not noticing one or two things. Can anyone help me please?


Half or more than half of the Quant section consists of DS questions. To get over 700, you need to be able to handle 700 level questions of DS as well. DS questions have a lot of traps and it pays to be aware of them. You can check out the Veritas DS book. It has excellent discussions on DS traps, common error areas and strategies. It's a must have if you are facing problems in DS but are relatively comfortable in PS because then you have DS format issues and need a book which addresses those specifically.
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Re: DS: number properties [#permalink]

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New post 22 Oct 2013, 21:50
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



Hi Karishma,

For option A, i took, X=10

then for x^2-2x+A>0, A can be positive , negative or eve zero.

100-20+A>0


how could you decide if A is positive??

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Re: DS: number properties [#permalink]

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New post 22 Oct 2013, 21:57
bsahil wrote:
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



Hi Karishma,

For option A, i took, X=10

then for x^2-2x+A>0, A can be positive , negative or eve zero.

100-20+A>0


how could you decide if A is positive??


X = 10 is fine but it doesn't help. We know that this inequality holds for all x. We need to plug in a value for x which tells us something about A. If we put x = 0, we are left with just A and that will tell us something about A. Just plugging in any value may not work; you have to look for a smart value.
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Re: DS: number properties [#permalink]

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New post 22 Oct 2013, 23:23
VeritasPrepKarishma wrote:

X = 10 is fine but it doesn't help. We know that this inequality holds for all x. We need to plug in a value for x which tells us something about A. If we put x = 0, we are left with just A and that will tell us something about A. Just plugging in any value may not work; you have to look for a smart value.


So for this question, no other value of x(except x=0,2) gives us any information about A. hence we take only that value which gives us explicit and fixed result for A. Am I correct??
Similarly we would do in other similar questions as well...??

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Re: DS: number properties [#permalink]

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New post 22 Oct 2013, 23:39
bsahil wrote:
VeritasPrepKarishma wrote:

X = 10 is fine but it doesn't help. We know that this inequality holds for all x. We need to plug in a value for x which tells us something about A. If we put x = 0, we are left with just A and that will tell us something about A. Just plugging in any value may not work; you have to look for a smart value.


So for this question, no other value of x(except x=0,2) gives us any information about A. hence we take only that value which gives us explicit and fixed result for A. Am I correct??
Similarly we would do in other similar questions as well...??


No, look, we know that \(x^2 - 2x + A > 0\) for all x. For every value of x, this inequality should be satisfied.

Put x = 0, you get A > 0
Put x = 1, you get \(1 - 2 + A > 0 i.e. A > 1\)
Put x = 10, you get A > -80

Now the point is that A should take a value such that all these conditions are satisfied. Say A can be 5. If A is 5, it is > 0, > 1 and > -80.

When I look at \(x^2 - 2x + A > 0\) given x can take any value, the first value that pops in my head to get a sense of A is x = 0. That provides exactly what I need. Had the question been whether A > 2, x = 0 would not have helped. I would have had to search a little for a pattern to see how the value of x changes. This question is made in a way that x = 0 helps immediately. Anyway, it is a good idea to try the value 0 in many circumstances. It simplifies things immensely and usually helps you eliminate a couple of options at least.
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Re: DS: number properties [#permalink]

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New post 23 Oct 2013, 21:24
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



if we put x=-3 in 1, then A can have -1 or -2 value also ??

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Re: DS: number properties [#permalink]

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New post 23 Oct 2013, 21:56
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ratanpandit wrote:
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



if we put x=-3 in 1, then A can have -1 or -2 value also ??


No, A can never have a value of -1 or -2. It must be greater than 1.
If we put x = -3, we get
X^2-2X+A > 0
A > -15
So this tells us that A must be greater than -15. Putting other values of x such as 0, 1, 2 etc tell us that A must be greater than 0 and A must be greater than 1 etc. Since this inequality holds for ALL values of x, A must be greater than 1 because a value greater than 1 will automatically be greater than -15 as well as 0. If we take a value of A such as -14, it will be greater than -15 but not greater than 0 or 1 hence the inequality will not hold for ALL value of x.
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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New post 21 Jul 2014, 08:30
Bunuel wrote:
noboru wrote:
Is A positive?

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


OA is A


Is \(A>0\)?

(1) \(x^2-2x+A\) is positive for all \(x\):

Quadratic expression \(x^2-2x+A\) is a function of of upward parabola (it's upward as coefficient of \(x^2\) is positive). We are told that this expression is positive for all \(x\) --> \(x^2-2x+A>0\), which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation \(x^2-2x+A=0\) has no real roots.

Quadratic equation to has no real roots discriminant must be negative --> \(D=2^2-4A=4-4A<0\) --> \(1-A<0\) --> \(A>1\).

Sufficient.

(2) \(Ax^2+1\) is positive for all \(x\):

\(Ax^2+1>0\) --> when \(A\geq0\) this expression is positive for all \(x\). So \(A\) can be zero too.

Not sufficient.

Answer: A.


Hello Bunnel
Can you kindly elaborate on an upward parabola (staying above X axis) having no real roots, please? When does a quadratic equation not have real roots?
What would be the case if it were a downward parabola not touching X axis?
Thank you!

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

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New post 21 Jul 2014, 10:09
Kconfused wrote:
Bunuel wrote:
noboru wrote:
Is A positive?

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


OA is A


Is \(A>0\)?

(1) \(x^2-2x+A\) is positive for all \(x\):

Quadratic expression \(x^2-2x+A\) is a function of of upward parabola (it's upward as coefficient of \(x^2\) is positive). We are told that this expression is positive for all \(x\) --> \(x^2-2x+A>0\), which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation \(x^2-2x+A=0\) has no real roots.

Quadratic equation to has no real roots discriminant must be negative --> \(D=2^2-4A=4-4A<0\) --> \(1-A<0\) --> \(A>1\).

Sufficient.

(2) \(Ax^2+1\) is positive for all \(x\):

\(Ax^2+1>0\) --> when \(A\geq0\) this expression is positive for all \(x\). So \(A\) can be zero too.

Not sufficient.

Answer: A.


Hello Bunnel
Can you kindly elaborate on an upward parabola (staying above X axis) having no real roots, please? When does a quadratic equation not have real roots?
What would be the case if it were a downward parabola not touching X axis?
Thank you!


Check Parabola chapter in Coordinate Geometry book: math-coordinate-geometry-87652.html

Hope it helps.
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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New post 21 Jul 2014, 10:28
Another way to approach this question is:

Statement 1: x^2-2x+A>0

Add and subtract 1 on the LHS

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

The first term is always>=0
Hence the second term,i.e. (A-1) must be greater than 0.
i.e. A>1

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

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New post 22 Jul 2014, 10:43
manu220194 wrote:
Another way to approach this question is:

Statement 1: x^2-2x+A>0

Add and subtract 1 on the LHS

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

The first term is always>=0
Hence the second term,i.e. (A-1) must be greater than 0.
i.e. A>1



Hi Manu,

YOu are missing one major fact in your solution
Given that (x-1)^2 +(A-1)>0 , but its not necessary that (A-1)>0. it might be that (x-1)^2 (positive) > (A-1) (negative) and that is the reason it is becoming positive.
Therefore, (A-1) can be negative also. [A-1<0 => A<1.... which does not gives us anything on the sufficiency of this statement.]

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

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New post 22 Jul 2014, 20:47
Thanks for pointing it out Aditi.
You are right! I missed it.

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

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New post 22 Jul 2014, 20:57
Found a better, sound solution compared to my previous, flawed one.

x^2-2x+A>0

A>-(x^2-2x)---------1.

Basically looked at the range of x^2-2x
Max(x^2-2x)=Infinity
Min(x^2-2x)=-1
Therefore, from 1.
A>-(infinity)
and A>-(-1), i.e. A>1
which is sufficient!

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

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New post 15 Aug 2015, 08:13
Bunuel wrote:
noboru wrote:
Is A positive?

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


OA is A


Is \(A>0\)?

(1) \(x^2-2x+A\) is positive for all \(x\):

Quadratic expression \(x^2-2x+A\) is a function of of upward parabola (it's upward as coefficient of \(x^2\) is positive). We are told that this expression is positive for all \(x\) --> \(x^2-2x+A>0\), which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation \(x^2-2x+A=0\) has no real roots.

Quadratic equation to has no real roots discriminant must be negative --> \(D=2^2-4A=4-4A<0\) --> \(1-A<0\) --> \(A>1\).

Sufficient.

(2) \(Ax^2+1\) is positive for all \(x\):

\(Ax^2+1>0\) --> when \(A\geq0\) this expression is positive for all \(x\). So \(A\) can be zero too.

Not sufficient.

Answer: A.


Hi Bunnel

How do we solve (2) using the discriminant method?

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

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New post 16 Aug 2015, 06:50
samichange wrote:


Hi Bunnel

How do we solve (2) using the discriminant method?

thanks


I have a different take as to why statement 2 is not sufficient

y = -2x^1+1 will give you y >0 (=0.5) for x = 0.5. Thus Bunuel 's explanation that A\(\geq\) 0 for y>0 for all x is not the complete explanation, IMHO.

Additionally, for A\(\geq\)0, y >0 for all x>0. Thus you get 2 different answers for "is A>0" based on these 2 cases and hence statement 2 is not sufficient.

samichange, using the discriminant method is not the best method for this statement.

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

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New post 16 Aug 2015, 07:03
Engr2012 wrote:
samichange wrote:


Hi Bunnel

How do we solve (2) using the discriminant method?

thanks


I have a different take as to why statement 2 is not sufficient

y = -2x^1+1 will give you y >0 (=0.5) for x = 0.5. Thus Bunuel 's explanation that A\(\geq\) 0 for y>0 for all x is not the complete explanation, IMHO.

Additionally, for A\(\geq\)0, y >0 for all x>0. Thus you get 2 different answers for "is A>0" based on these 2 cases and hence statement 2 is not sufficient.

samichange, using the discriminant method is not the best method for this statement.


Hi

Thanks for your analysis but what I understood is the following-

Consider x =0 and then whatever be the value of a ( + , - or 0 ), it does not matter because y = 0 + 1 ---> y > 0 and hence B is insufficient as a can assume any value.
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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New post 16 Aug 2015, 07:06
I also think that the discriminant method is not a great method because then we have to necessarily assume that y = ax^2+1 represents a quadratic equation when y can be a straight line y =1 for x =0 or a=0 or x=a=0.
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Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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samichange wrote:
Engr2012 wrote:
samichange wrote:


Hi Bunnel

How do we solve (2) using the discriminant method?

thanks


I have a different take as to why statement 2 is not sufficient

y = -2x^1+1 will give you y >0 (=0.5) for x = 0.5. Thus Bunuel 's explanation that A\(\geq\) 0 for y>0 for all x is not the complete explanation, IMHO.

Additionally, for A\(\geq\)0, y >0 for all x>0. Thus you get 2 different answers for "is A>0" based on these 2 cases and hence statement 2 is not sufficient.

samichange, using the discriminant method is not the best method for this statement.


Hi

Thanks for your analysis but what I understood is the following-

Consider x =0 and then whatever be the value of a ( + , - or 0 ), it does not matter because y = 0 + 1 ---> y > 0 and hence B is insufficient as a can assume any value.


Yes, you are correct about your explanation of why B is not sufficient and why discriminant method is not the wisest choice for this statement. Statement 1 was straightforward with the discriminant method as it would have remained a quadratic equation no matter what the value of A was but statement 2 will give an equation of a line if A = 0.

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

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New post 06 Dec 2015, 20:50
Is A > 0?

St 1: (x^2 - 2*x) + A > 0 for all values of x.

if (x^2 - 2*x) <= 0 for any value of x then A > 0.

x^2 - 2*x = x(x-2) and on the number line for any x = [0,2] ==> x(x-2) <= 0, otherwise x(x-2) is positive. A + x(x-2) is always positive therefore A is positive.

Suff

St 2: A*x^2+1 > 0 for all x

A > -1/x^2

Therefore A can be < or > 0.

Insuff.

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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is   [#permalink] 06 Dec 2015, 20:50

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