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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 have 1 variable (m) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.
Condition 1) tells us that the difference between the two roots of x^2 + (1 + m)x + 20 = 0 is 1.
We can assume p and p+1 are the roots of the equation x^2 + (1 + m)x + 20 = 0.
We have (x - p)(x - (p + 1)) = x^2 - (2p + 1)x + p(p + 1) = x^2 + (1 + m)x + 20.
Then, we have 1 + m = -2p – 1 or m = -2p – 2. We also have p(p + 1) = 20 or p^2 + p - 20 = (p - 4)(p + 5) = 0.
Thus p = 4 or p = -5, which we can substitute into the first equation giving us m = -2p – 2 = -2·4 – 2 = -10, or m = -2p – 2 = -2·(-5) – 2 = 8 .
Then we have m = -10 or m = 8.
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) tells us that m > 0, from which we cannot determine the value of m. For example, m can be 2 or 3.
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) together tell us that the answer, m = 8 is unique, and both conditions are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) and 2) together are sufficient.
Therefore, C is the answer.
Answer: C
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.
