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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.
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Since we can figure s and t out if we know the value of a, we have 1 variable 1 and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.
Condition 1)
Since two roots are equal, its discriminant is 0. Then, we have (-4a)^2 – 4·2(a^2 + 1) = 0 , 16a^2 – 8a^2 – 8 = 0, 8a^2 – 8 = 0, 8(a^2 - 1) = 0, 8(a + 1)(a - 1) = 0.
Thus, we have a = -1 or a = 1.
The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition 2):
a = s = t = 1 and a = s = t = -1 are solutions to the question.
The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Conditions 1) & 2)
We have two solutions, a = s = t = 1 and a = s = t = -1 even when we consider both conditions together.
The answer is not unique, and the conditions are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Therefore, E is the answer.
Answer: E
If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations,” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
