Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (\(x\) and \(y\)) and 0 equations, C is most likely to be the answer. So, we should consider 1) & 2) first.

Conditions 1) & 2):

\(x^y = (y^2)^y = y^{2y}\)

\(y^x = y^{y^2}\)

Since \(y > 2\), we have \(y^2 > 2y\) and \(y^{2y} < y^{y^2}\).

Thus both conditions together are sufficient.

Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT(Common Mistake Type) 4(A).

Condition 1)

\(x = 9, y = 3 ⇒ 9^3 = 3^6 < 3^9\) : Yes

\(x = 1, y = 1 ⇒ 1^1 = 1^1\) : No

The condition 1) only is not sufficient.

Condition 2)

\(x = 1, y = 4 ⇒ 1^2 < 4^1\) : Yes

\(x = -1, y = 4 => (-1)^4 = 1 > 4^{-1} = \frac{1}{4}\) : No

The condition 2) only is not sufficient.

Therefore, C is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.

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