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06 Jul 2018, 01:43
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
26% (02:20) correct
74%
(02:25)
wrong
based on 121
sessions
History
Date
Time
Result
Not Attempted Yet
In the x-y plane, \(y = f(x) = ax^2+bx+c\) passes through \((1,0)\) and \((2,0)\). Is \(f(3)>0?\)
(1) \(a > 0\)
(2) \(f(0) > 0\)
(1) \(a > 0\)
(2) \(f(0) > 0\)
06 Jul 2018, 04:08
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MathRevolution
\(f(x) = ax^2+bx+c\) will be a U-shaped parabola if a>0.
\(f(x) = ax^2+bx+c\) will be a ∩-shaped parabola if a<0.
Since \(f(x) = ax^2+bx+c\) has x-intercepts at (1, 0) and (2, 0), \(f(3)>0\) if the parabola is U-shaped.
Question stem, rephrased:
Is the parabola U-shaped?
Statement 1:
Since a>0, the parabola is U-shaped.
SUFFICIENT.
Statement 2:
x=0 is to the LEFT of the x-intercept at (1, 0).
Since x=0 yields a POSITIVE y-value, the parabola must be U-shaped.
SUFFICIENT.
08 Jul 2018, 04:17
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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.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
Since f(x) passes through (1,0) and (2,0), 1 and 2 are roots of f(x) and we have f(x) = a(x-1)(x-2). If f(x) is concave up, then f(3) > 0. Thus, knowing that f(x) is concave up will allow us to answer the question. In addition, since we have 3 variables (a, b, c) and 2 conditions, D is most likely to be the answer.
Condition 1)
If a > 0, then f(x) is concave up.
Thus, condition 1) is sufficient.
Condition 2)
Since 1 and 2 are roots of f(x) = 0, f(0) > 0 implies that f(x) is concave up.
Thus, condition 2) is sufficient, too.
Therefore, D is the answer.
Answer: D
Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
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.
The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
Since f(x) passes through (1,0) and (2,0), 1 and 2 are roots of f(x) and we have f(x) = a(x-1)(x-2). If f(x) is concave up, then f(3) > 0. Thus, knowing that f(x) is concave up will allow us to answer the question. In addition, since we have 3 variables (a, b, c) and 2 conditions, D is most likely to be the answer.
Condition 1)
If a > 0, then f(x) is concave up.
Thus, condition 1) is sufficient.
Condition 2)
Since 1 and 2 are roots of f(x) = 0, f(0) > 0 implies that f(x) is concave up.
Thus, condition 2) is sufficient, too.
Therefore, D is the answer.
Answer: D
Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
