Hi Anindya,
First, we want to simplify the question so we know exactly what we're looking for.
x^Y * y^(-x) =1
When we have a negative exponent, we can express it as the reciprocal of the positive exponent, so turn y^(-x) into 1/(y^x)
(x^y)*(1/(y^x)) = 1 then becomes (x^y)/(y^x) = 1
Cross multiply, and you get the question: is x^y = y^x ?
So, in what situations is x^y equal to y^x? Clearly if x and y are the same value, then we'll get a "yes" answer. But think hard to see if there are other scenarios - what about 2^4 and 4^2?
Let's look at the statements to see if we can definitely prove whether x^y = y^x.
(1) x^x > y
The best way to approach DS statements is always to try to prove insufficiency. So taking the statement, let's find an example where x^y and y^x are equal, and one where they're not.
If we say that x and y are both equal to 3, for example, then 3^3 > 3, so it satisfies the statement. 3^3 is equal to 3^3, so we'd get a "yes" answer to our question.
But, if we say that x = 3 and y = 2, then 3^3 > 2, and again we satisfy the statement. When we plug those values into the question, though, 3^2 is not equal to 2^3, so we get a "no" answer. Statement 1 is not sufficient, so we can eliminate A and D.
(2) x > y^y
Again, let's try to prove insufficiency by testing values. We can say that x = 5 and y = 2, which satisfies the statement, because 5 > 2^2. When we look at the question, 5^2 is not equal to 2^5, so we get a "no" answer.
So, can we get a "yes" answer to the question? Can we think of a situation where x > y^y, and x^y = y^x? If x > y^y, it eliminates the possibility that x and y are equal to each other. (Try it out: is 2 > 2^2? No. Is -3 > (-3)^(-3)? No. This won't be true for any integers). It also eliminates the possibility of x = 2 and y = 4, or vice versa, since neither of those would satisfy the statement.
Statement 2 eliminates both of the possibilities that would have given us a "yes" answer to the question, so it tells us that the answer must be "no." Statement 2 is therefore sufficient.
The answer is B.
