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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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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 1 variable (k) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.
Condition 1) tells us that k is a positive integer, from which we cannot determine the unique set of A. If k = 3, then x^2 - 2(k - 1)x + 4 = x^2 – 2(3 – 1)x + 4 = x^2 - 4x + 4 = (x - 2)^2 = 0 and A = {2}. However, if k = 1, then x^2 - 2(k - 1)x + 4 = x^2 - 2(1 – 1) + 4 = x^2 + 4 = 0, which does not have a root and A is an empty set.
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 the number of elements in A is 1, from which we get that the discriminant of the quadratic equation is 0.
Since the number of roots of the equation is 1, its discriminant
(b^2 – 4ac)
= [2(k – 1)]^2 - 4·1·4
= 4(k - 1)^2 - 16
= 4(k^2 - 2k + 1) - 4·4
= 4(k^2 – 2k + 1 - ) 4 (taking out a common factor of 4)
= 4(k^2 - 2k - 3)
= 4(k + 1)(k - 3) is zero.
Then, we have 4(k + 1)(k - 3) = 0 and k = -1 or k = 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)
Then only k = 3 is the unique answer.
The answer is unique, and both conditions together 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.
