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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 3 variables (x, y, and z) and 0 equations, E 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)
When we replace x in the equation ax^2 + bx + c = 0 with 2 from condition 1), we have 4a + 2b + c = 0.
When we substitute x and y in the equation y = ax^2 with -1 and 2, respectively, we have 2 = a(-1)^2, and a = 2.
When we substitute x and y in the equation y = -bx - c with -1 and 2, respectively, we have 2 = -b(-1) – c, and b – c = 2.
If we replace a in the equation 4a + 2b + c = 0 with 2, we have 4(2) +2b +c = 0, and 2b + c = -8.
When we add the last two equations, we get b – c + 2b + c = 2 – 8, 3b = -6 or b = -2.
If we replace b in the equation b – c = 2 with -2, we have -2 – c = 2, -c = 4, and c = -4.
Thus, we have a + b + c = 2 +(-2) + (-4) = -4.
The answer is unique, and both conditions combined are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Therefore, C is the correct answer.
Answer: C
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
