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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 the question includes 2 variables (x and y) and no equation, C is most likely to be the answer. Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT 4(A).

Conditions 1) & 2)

Since \(x^2 + y^2\) has a remainder of 2 when it is divided by 4, \(x^2 + y^2\) must be even. Since x is odd, \(x^2\) is odd and so \(y^2\) must also be odd. Therefore, y is odd, and \(x + y\) is even. The answer is ‘yes’.

Condition 1)

Since we don’t know whether y is even or odd, this is not sufficient.

Condition 2)

The condition tells us that \(x^2+y^2=4k+2=2(2k+1)\) is even. Since\(x^2+y^2=(x+y)^2-2xy\), and \(2xy\) is even, this implies that \((x+y)^2\) is also even. But this can only happen if x+y is even. So, the answer is ‘yes’.

This condition is sufficient.

Therefore, the answer is B.

Normally, in problems which require 2 or more additional 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).

Answer: B

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