A

[goyalsau]

Is A positive?

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

[Rahul@gurome]

Solution:
x^2 -2x+A = (x-1)^2 - 1 +A is always positive.
Now (x-1)^2 is always 0 or positive.
Let (x-1)^2 be 4 and A be -2. So (x-1)^2 - 1 +A = 4 -1 - 2 = 1 is positive and A = -2 is negative.
Next let (x-1)^2 be 1 and A be 2. So (x-1)^2 - 1 + A = 1 - 1 + 2 = 2 is positive and A = 2 is positive.
Since nothing definite can be said about A, (1) alone is not sufficient.
Next consider (2) alone.
Let x^2 be 1 and A be -1/2. So Ax^2+1 is -1/2 + 1 = ½ is positive and A = -1/2 is negative.
Next let x^2 be 1 and A be 2. So Ax^2+1 is 2 + 1 = 3 is positive and A = 2 is positive.
Again, nothing definite can be said and so (2) alone is not sufficient.
Next combine both statements together and check.
Let x= 3 and A = 1.
So x^2 -2x+A = 9 - 6 +1 = 4 is positive, Ax^2+1 = 9 +1 = 10 is positive and A = 1 is positive.
Next let x = -3 and A = -1/10.
So x^2-2x+A = 9 + 6 - 1/10 is positive, Ax^2+1 = -9/10 + 1 = 1/10 is positive and A = -1/10 is negative.
Again nothing definite can be said from (1) and (2) combined.

The correct answer is (E).

[diebeatsthegmat]

Is A positive?

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

is that C the answer?

1/ x^2-2x+a>0
if x=1/2 and A =4 so 1/4-1+4>0 yes
if x=3 and A=-2 so 9-6-3=1>0 no
insufficient
2/ax^2+1> so
x=1/2 A=-3 > -3/4+1=1/4>0
if x=3 and A=1 so insufficient
1+2 / x must be integer so its sufficient
C?

[diebeatsthegmat]

Solution:
x^2 -2x+A = (x-1)^2 - 1 +A is always positive.
Now (x-1)^2 is always 0 or positive.
Let (x-1)^2 be 4 and A be -2. So (x-1)^2 - 1 +A = 4 -1 - 2 = 1 is positive and A = -2 is negative.
Next let (x-1)^2 be 1 and A be 2. So (x-1)^2 - 1 + A = 1 - 1 + 2 = 2 is positive and A = 2 is positive.
Since nothing definite can be said about A, (1) alone is not sufficient.
Next consider (2) alone.
Let x^2 be 1 and A be -1/2. So Ax^2+1 is -1/2 + 1 = ½ is positive and A = -1/2 is negative.
Next let x^2 be 1 and A be 2. So Ax^2+1 is 2 + 1 = 3 is positive and A = 2 is positive.
Again, nothing definite can be said and so (2) alone is not sufficient.
Next combine both statements together and check.
Let x= 3 and A = 1.
So x^2 -2x+A = 9 - 6 +1 = 4 is positive, Ax^2+1 = 9 +1 = 10 is positive and A = 1 is positive.
Next let x = -3 and A = -1/10.
So x^2-2x+A = 9 + 6 - 1/10 is positive, Ax^2+1 = -9/10 + 1 = 1/10 is positive and A = -1/10 is negative.
Again nothing definite can be said from (1) and (2) combined.

The correct answer is (E).

awww understood... thanks...

[goyalsau]

Solution:
x^2 -2x+A = (x-1)^2 - 1 +A is always positive.
Now (x-1)^2 is always 0 or positive.
Let (x-1)^2 be 4 and A be -2. So (x-1)^2 - 1 +A = 4 -1 - 2 = 1 is positive and A = -2 is negative.
Next let (x-1)^2 be 1 and A be 2. So (x-1)^2 - 1 + A = 1 - 1 + 2 = 2 is positive and A = 2 is positive.
Since nothing definite can be said about A, (1) alone is not sufficient.
Next consider (2) alone.
Let x^2 be 1 and A be -1/2. So Ax^2+1 is -1/2 + 1 = ½ is positive and A = -1/2 is negative.
Next let x^2 be 1 and A be 2. So Ax^2+1 is 2 + 1 = 3 is positive and A = 2 is positive.
Again, nothing definite can be said and so (2) alone is not sufficient.
Next combine both statements together and check.
Let x= 3 and A = 1.
So x^2 -2x+A = 9 - 6 +1 = 4 is positive, Ax^2+1 = 9 +1 = 10 is positive and A = 1 is positive.
Next let x = -3 and A = -1/10.
So x^2-2x+A = 9 + 6 - 1/10 is positive, Ax^2+1 = -9/10 + 1 = 1/10 is positive and A = -1/10 is negative.
Again nothing definite can be said from (1) and (2) combined.

The correct answer is (E).

Excellent work Rahul
can you please tell me how to assume these values for A and X,
Is there any relation between X and A for assuming these values.

[Ian Stewart]

Solution:
x^2 -2x+A = (x-1)^2 - 1 +A is always positive.
Now (x-1)^2 is always 0 or positive.
Let (x-1)^2 be 4 and A be -2. So (x-1)^2 - 1 +A = 4 -1 - 2 = 1 is positive and A = -2 is negative.
Next let (x-1)^2 be 1 and A be 2. So (x-1)^2 - 1 + A = 1 - 1 + 2 = 2 is positive and A = 2 is positive.
Since nothing definite can be said about A, (1) alone is not sufficient.
Next consider (2) alone.
Let x^2 be 1 and A be -1/2. So Ax^2+1 is -1/2 + 1 = ½ is positive and A = -1/2 is negative.
Next let x^2 be 1 and A be 2. So Ax^2+1 is 2 + 1 = 3 is positive and A = 2 is positive.
Again, nothing definite can be said and so (2) alone is not sufficient.
Next combine both statements together and check.
Let x= 3 and A = 1.
So x^2 -2x+A = 9 - 6 +1 = 4 is positive, Ax^2+1 = 9 +1 = 10 is positive and A = 1 is positive.
Next let x = -3 and A = -1/10.
So x^2-2x+A = 9 + 6 - 1/10 is positive, Ax^2+1 = -9/10 + 1 = 1/10 is positive and A = -1/10 is negative.
Again nothing definite can be said from (1) and (2) combined.

The correct answer is (E).

The above is backwards; you're assuming the answer to the question is 'no, and you're proving whether the statements are true or false. The statements can't be false; they're facts. We have to assume the statements are true, and try to answer the question.

From Statement 1, we learn that x^2 - 2x + A is positive for *all* values of x, not just for *one* value of x. If it's positive for every value of x, it's certainly positive when x = 0, so plugging in x = 0, we learn that A is positive, and Statement 1 is sufficient.

For Statement 2, Ax^2 + 1 is positive for all x. Certainly A could be positive, but A can also be zero, so Statement 2 is not sufficient.

The answer is A.

[goyalsau]

The above is backwards; you're assuming the answer to the question is 'no, and you're proving whether the statements are true or false. The statements can't be false; they're facts. We have to assume the statements are true, and try to answer the question.

From Statement 1, we learn that x^2 - 2x + A is positive for *all* values of x, not just for *one* value of x. If it's positive for every value of x, it's certainly positive when x = 0, so plugging in x = 0, we learn that A is positive, and Statement 1 is sufficient.

For Statement 2, Ax^2 + 1 is positive for all x. Certainly A could be positive, but A can also be zero, so Statement 2 is not sufficient.

The answer is A


OA is A, But i am not able to understand the reasoning behind the answer.

If one statement is sufficient , It should always give the same answer when we put in different values, Always Yes or always No,

But
x^2 - 2x + A I definitely agree with you when x = 0 , A is +ve
But if X = 10 We will left with
80 + A In this case we can put in the value of A from -1 to -79 end result of the equation will be +ve , if we put 0 then also answer it will be +ve , And off course we can put A as +ve as well.

Can you please very specifically mention how you can ignore this possibility,

I m totally confused, I really don't know how to approach DS now.

[this_time_i_will]

i wonder how stmt. I is sufficient!

I: x^2+A>2x
so, x = -2 ; A = 1 & x = -2; A=-1. In both cases the inequality holds but A is positive in former and negative in latter.

[goyalsau]

i wonder how stmt. I is sufficient!

I: x^2+A>2x
so, x = -2 ; A = 1 & x = -2; A=-1. In both cases the inequality holds but A is positive in former and negative in latter.

OA is Also A,
So there must be something that Ian can make it clear.

[Ian Stewart]

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

You are both misinterpreting Statement 1. Statement 1 tells us that x^2 - 2x + A is positive for **EVERY** value of x. You both seem to be interpreting Statement 1 to mean "x^2 - 2x + A is positive for *some* value of x", but that's not what it says. So here:

But
x^2 - 2x + A I definitely agree with you when x = 0 , A is +ve
But if X = 10 We will left with
80 + A In this case we can put in the value of A from -1 to -79 end result of the equation will be +ve , if we put 0 then also answer it will be +ve , And off course we can put A as +ve as well.

Can you please very specifically mention how you can ignore this possibility,

we know that x^2 - 2x + A > 0 must be true when x = 10, since it's true for every value of x. So by plugging in x = 10, you learn that A > -80. But, from Statement 1, x^2 - 2x + A > 0 must be true not *only* when x = 10, but also when x = 3.77, or when x = -250, or when x = 0.

So while you learn something about A by plugging in x = 10, you certainly aren't finished analyzing Statement 1, since you haven't considered all of the other possible values of x.

i wonder how stmt. I is sufficient!

I: x^2+A>2x
so, x = -2 ; A = 1 & x = -2; A=-1. In both cases the inequality holds but A is positive in former and negative in latter.

And again, by plugging in x = -2, you learn that A > -8. But the inequality in Statement 1 needs to be true for every other value of x you can try as well, not only for x = -2.

[goyalsau]

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

You are both misinterpreting Statement 1. Statement 1 tells us that x^2 - 2x + A is positive for **EVERY** value of x. You both seem to be interpreting Statement 1 to mean "x^2 - 2x + A is positive for *some* value of x", but that's not what it says. So here:

But
x^2 - 2x + A I definitely agree with you when x = 0 , A is +ve
But if X = 10 We will left with
80 + A In this case we can put in the value of A from -1 to -79 end result of the equation will be +ve , if we put 0 then also answer it will be +ve , And off course we can put A as +ve as well.

Can you please very specifically mention how you can ignore this possibility,

we know that x^2 - 2x + A > 0 must be true when x = 10, since it's true for every value of x. So by plugging in x = 10, you learn that A > -80. But, from Statement 1, x^2 - 2x + A > 0 must be true not *only* when x = 10, but also when x = 3.77, or when x = -250, or when x = 0.

So while you learn something about A by plugging in x = 10, you certainly aren't finished analyzing Statement 1, since you haven't considered all of the other possible values of x.

i wonder how stmt. I is sufficient!

I: x^2+A>2x
so, x = -2 ; A = 1 & x = -2; A=-1. In both cases the inequality holds but A is positive in former and negative in latter.

And again, by plugging in x = -2, you learn that A > -8. But the inequality in Statement 1 needs to be true for every other value of x you can try as well, not only for x = -2.

I think i got it now,
What i was doing , I was trying to prove statement as incorrect
And this is what i can't do, Because as you said they are facts.

Thanks Ian thanks a lot........

[GMATGuruNY]

Is A positive?

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

Here's another approach:

When examining statement 1, we need to answer the following question:

What values of A will guarantee that x^2-2x+A > 0 for all values of x?

The minimum value of x^2-2x is -1, when x = 1. (See footnote below.)
Thus, to guarantee that x^2-2x+A > 0 for all values of x, A must be greater than 1.
Sufficient.

When examining statement 2, we need to answer the following question:

What values of A will guarantee that Ax^2+1> 0 for all values of x?

A=0 works, because then Ax^2+1 = 1, which is positive for all values of x.
A=1 works, because then Ax^2+1 = x^2+1, which is positive for all values of x.
Since we can't tell whether A must be positive, insufficient.

The correct answer is A.

Footnote:
Plug any value other than x=1 into x^2-2x, and you'll get a result bigger than -1. Thus, -1 is the minimum value of x^2-2x. Or to determine what value of x will yield the minimum, set the derivative equal to 0:

2x-2 = 0
x = 1.
Plugging x=1 into x^2-2x, we see that its minimum value is 1^2 - 2*1 = -1.

This latter approach is helpful but beyond the scope of the GMAT.