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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.
Since we have 2 variables (x and y) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2) together give us:
x = (1 + A) ○ (1 - A) = [1 / (1 + A + 1 - A)] – [1 / (1 + A)] = [1/2] – [1 / (1 + A)] = [(1 + A) / 2(1 + A)] – [2 / 2(1 + A)] = [(1 + A - 2) / 2(1 + A)] = [(A - 1) / 2(1 + A)].
y = (1 - A) ○ (1 + A) = [1 / (1 – A + 1 + A)] – [1 / (1 - A)] = [1/2] – [1 / (1 - A)] = [(1 - A) / 2(1 - A)] – [2 / 2(1 - A)] = [(1 – A - 2) / 2(1 - A)] = -(A + 1) / 2(1 - A).
Then, we have xy = [(A - 1) / 2(1 + A)] · [-(A + 1) / 2(1 - A)] = [(A – 1)·(-A – 1)] / [4·(1 + A)·(1 – A)] = [-A^2 – A + A + 1] / [4·(1 – A + A - A^2] = [-A^2 + 1] / [4·(1 – A^2] = [1 – A^2] / [4·(1 - A^2] = ¼.
The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) & 2) together are sufficient.
Therefore, C is the correct answer.
Answer: C
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.
