Solution: 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.
Visit
https://www.mathrevolution.com/gmat/lesson for details.
Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find whether \(x^2-y^2\) is an integer, whether (x-y)(x+y) is an integer
Thus, let’s look at condition (1), it tells us that x = y, from which we get x - y = 0 or (x-y)(x+y)=0 => (x+y)=0, which is an integer.
And we get yes an answer, the answer is unique, yes, the condition is sufficient according to Common Mistake Type 1 which states that the answer should be unique Yes or a NO.
Condition (2) tells us that y = 2, from which we cannot determine whether \(x^2-y^2\) is an integer.
For example, if x = 1 and y = 2, then we get \(x^2-y^2\) = \(1^2 - 2^2\)=1 - 4 = -3 which is an integer and we get yes as an answer.
However, if x = 0.1 and y = 2, then we get \(x^2 - y^2\)= \(0.1^2 - 2^2\) = 0.01 - 4=-3.99 which is not an integer and we get no as an answer.
The answer is not unique, yes and no, and condition (1) alone is not sufficient according to Common Mistake Type 1 which states that the answer should be unique Yes or a NO if we get both yes and no as an answer, it is not sufficient.
Condition (1) alone is sufficient
Therefore, A is the correct answer
Answer: A
