Challenging Question

[Mo2men]

Is A positive?

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

OA: A

Source: GMAT Club tests

[GMATGuruNY]

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Hi Mo2men,

While it might be tempting to approach this prompt in an overly-technical way, I'm going to focus instead on pattern matching and TESTing Values, which will likely be easier to understand.

We're asked: Is A positive? This is a YES/NO question.

Fact 1: X^2 - 2X + A = positive FOR ALL VALUES OF X

This means that we need a value for A that will make the above result positive NO MATTER WHAT X IS.....

In this type of situation, I go looking at the possibilities:

If X = 0, then would have to be something POSITIVE
If X = 1, then A would have to be >1
If X = 2, then A would have to be something POSITIVE
If X = anything negative, then the A could be positive OR negative
Etc.
With this information, the ONLY way to account for all POSSIBLE values of X is for A > 1
This means that A MUST BE positive and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

Fact 2: A(X^2) + 1 = positive FOR ALL VALUES OF X

Again, I'd look at the possibilities...

Since (X^2) = 0 or a POSITIVE, A would only need to be greater than OR EQUAL TO 0.

If A = 0, then the answer is NO.
If A > 0, then the answer is YES.
Fact 2 is INSUFFICIENT.

Final Answer: A

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