Algebraically we can see why the answer is A. If we know that \(a = b^2\), we can substitute for a in the question. Bear in mind what we're doing - we're rephrasing the question, or in other words, finding out under what conditions the answer to the question will be 'yes'.
Is \(a^b > b^a\) ?
Is \((b^2)^b > b^{(b^2)}\) ?
Is \(b^{2b} > b^{(b^2)}\) ?
and since our base is greater than 1, this inequality will only be true if the exponent on the left is larger than the exponent on the right, so our question becomes:
Is \(2b > b^2\) ?
and since b is positive, we can divide by b without worrying about reversing the inequality:
Is \(2 > b\) ?
So the answer to the question is only 'yes' when \(b < 2\) is true. Since the question tells us this is absolutely not true (it tells us b is an integer greater than 1, so we know b
> 2), we know the answer to the question must be 'no'.
One final comment:
soumyajit_nayak wrote:
Statement 2: b>2
No information about a. So, statement 2 is insufficient to answer.
One needs to be careful using logic like this in GMAT DS questions. It's very possible for a statement that doesn't mention
a here to be sufficient, and the trickiest real GMAT DS questions are often ones with statements that, at first glance, don't appear to have a chance to be sufficient, but which turn out to be. I can quickly make up a simple example, but far more devious examples are possible:
If
k and
z are positive integers, is \(k^z \geq z^k\)?
1. \(z = 1\)
2. \(k > z\)
Here if you thought "statement 1 doesn't mention k, so it cannot be sufficient" you'd be dismissing Statement 1 too quickly. If we know z=1 from Statement 1, then substituting, the question becomes "is \(k^1 \geq 1^k\)?", so the question becomes "Is \(k \geq 1\)?" which we know is true. So Statement 1 is sufficient. Statement 2 is not (you can get a 'yes' answer whenever z=1, but a 'no' answer when k=4 and z=3, say), so A would be the answer here.
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