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If x=1/2, then x^2-2x+A = -3/4 + A. So A -> positive if x^2-2x+A is positive.
If x=5, then x^2-2x+A = 15+A. So A -> positive or A is negative but bigger than -15. However, since we're saying that it must be all x, A must be positive otherwise fractional x wouldn't work. Sufficient.

St2:
A*x^2 + 1 is positive for all x

If x=1/2, then A*x^2 + 1 = A/4 + 1. A can be negative or positive and A*x^2 + 1 will be positive.

If x=2, then A*x^2 + 1 = 4a + 1. A has to be positive if A*x^2 + 1 is positive. Since we want A*x^2 + 1 to be positive for all x, then A has to be positive if not integer values of x won't work. Sufficient.

If x=1/2, then x^2-2x+A = -3/4 + A. So A -> positive if x^2-2x+A is positive. If x=5, then x^2-2x+A = 15+A. So A -> positive or A is negative but bigger than -15. However, since we're saying that it must be all x, A must be positive otherwise fractional x wouldn't work. Sufficient.

St2: A*x^2 + 1 is positive for all x

If x=1/2, then A*x^2 + 1 = A/4 + 1. A can be negative or positive and A*x^2 + 1 will be positive.

If x=2, then A*x^2 + 1 = 4a + 1. A has to be positive if A*x^2 + 1 is positive. Since we want A*x^2 + 1 to be positive for all x, then A has to be positive if not integer values of x won't work. Sufficient.

Ans D

I disagree with D.

From stmt 1 we know A > 2x. This doesnt mean that A is positive or negative.
From Stmt 2 we know that A > -1/x^2. So no clue here too

1) x^2 - 2x +A is positive for all x 2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

i think it is E

1) A could be positive or negative
ex. x = 4, A = -5 -> 4^2 - 2(4) - 5 = 3
stmt still true

ex. x = 4, A = 5 -> 4^2 - 2(4) + 5 = 13
stmt still true

2) A could be either again since does not specify whether int or not
x^2 is always positive but A can be neg (if the product for A* x^2 is -1/2 for instance.. stmt still holds true) or A can positive and yield a positive answer

1) x^2 - 2x +A is positive for all x 2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

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

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0
so it is reduced to (x - 1)^2 + A – 1 > 0.
now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

1) x^2 - 2x +A is positive for all x 2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

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

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0 so it is reduced to (x - 1)^2 + A – 1 > 0. now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.

Indeed a good approach Fistail.

But how about putting the value of x = -1 and obtaining the possible values of A as I mentioned in my above earlier post?

1) x^2 - 2x +A is positive for all x 2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

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

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0 so it is reduced to (x - 1)^2 + A – 1 > 0. now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.

Indeed a good approach Fistail.

But how about putting the value of x = -1 and obtaining the possible values of A as I mentioned in my above earlier post?

Writing again:

ST1: x^2 - 2x +A is positive for all x

Let's say x = -1

Then, 1 + 2 + A > 0 => 3 + A > 0 => A > (-3)

So A may be -2 , -1 , 0 or > 0

Hence A may be positive or negative.

- Brajesh

I again looked at your approach.

As per you:

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0 so it is reduced to (x - 1)^2 + A – 1 > 0. now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff. Why are you assuming only the case when (x-1)^2 is equal to 0?

Well, (x -1)^2 will always be >= 0 but it does not mean that "A – 1" has to be positive for all the cases.
Imagine, (x -1)^2 is equal to 4 (by taking x = 3 ) , in this case, (A-1) can be (-3) i.e A can be (-2) and still the whole inequality will be intact.

1) x^2 - 2x +A is positive for all x 2) A*x^2 + 1 is positive for all x

Pls. explain. Thanks.

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

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0 so it is reduced to (x - 1)^2 + A – 1 > 0. now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff.

2: A*x^2 + 1 is positive for all x.

it is clearly insufficient because x^2 is 0, A could be +ve or -ve and the expression still is +ve..

so A works here.

Indeed a good approach Fistail.

But how about putting the value of x = -1 and obtaining the possible values of A as I mentioned in my above earlier post?

Writing again:

ST1: x^2 - 2x +A is positive for all x

Let's say x = -1

Then, 1 + 2 + A > 0 => 3 + A > 0 => A > (-3)

So A may be -2 , -1 , 0 or > 0

Hence A may be positive or negative.

- Brajesh

I again looked at your approach.

As per you:

refrese the inequality as : x^2 - 2x + 1 + A -1 > 0 so it is reduced to (x - 1)^2 + A – 1 > 0. now (x -1)^2 can be 0 or grater than 0. if it is 0, A - 1 has to be +ve and to be so, A has to be grater than 1. so suff. Why are you assuming only the case when (x-1)^2 is equal to 0?

Well, (x -1)^2 will always be >= 0 but it does not mean that "A – 1" has to be positive for all the cases. Imagine, (x -1)^2 is equal to 4 (by taking x = 3 ) , in this case, (A-1) can be (-3) i.e A can be (-2) and still the whole inequality will be intact.

Please let me know your opinion.

- Brajesh

The OA is A. But it doesn't make sense. I'm having the same problem understanding the solution as Brijesh (Above). Any thoughts?

The catch in this question is that A is CONSTATNT not a variable. So for all x, A will have the same value.

Now read it as - Find the value of A (+/-) for which x^2 - 2x + A > 0 is true for all x.
Paraphrase this (x-1)^2 + A-1 > 0
The minimum value of (x-1)^2 is zero. So A > 1.
So A> 1 satisfies the expression for all x.
Hence SUFF.

Stmt2: Ax^2 + 1 > 0
If we take A -ve, the above expression will be true for some cases and false for some cases depending on value of x. To make this true for all x, A should be +ve or zero.
So INSUFF.

My answer is 'A'

Last edited by vshaunak@gmail.com on 21 Sep 2007, 03:02, edited 1 time in total.

The catch in this question is that A is CONSTATNT not a variable. So for all x, A will have the same value.

Now read it as - Find the value of A (+/-) for which x^2 - 2x + A > 0 is true for all x. Paraphrase this (x-1)^2 + A-1 > 0 The minimum value of (x-1)^2 is zero. So A > 1. So A> 1 satisfies the expression for all x. Hence SUFF.

Stmt2: Ax^2 + 1 > 0 If we take A -ve, the above expression will be true for some cases and false for some cases depending on value of x. To make this true for all x, A should be +ve. So SUFF.

My answer is 'D'

I think OA is correct as A .
For Stmtn 1 - same as marked as blue .

For Stmtn 2 - you can never determine whether A is +ve or -ve

Ax^2 + 1 = 0.5 => A = -ve
Ax^2 +1 = 2 => A = +ve
So stmtn is not suff .