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Updated on: 21 Mar 2021, 03:00
Originally posted by MathRevolution on 12 Jun 2020, 02:17.
Last edited by MathRevolution on 21 Mar 2021, 03:00, edited 1 time in total.
Last edited by MathRevolution on 21 Mar 2021, 03:00, edited 1 time in total.
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MathRevolution
=>
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.
Visit https://www.mathrevolution.com/gmat/lesson for details.
Since a function has many variables to determine, 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) together give us:
When we substitute \(x\) in the equation \(f(x)+2f(\frac{1}{x})=3x\) by \(\frac{1}{x},\) we have \(f(\frac{1}{x}) + 2f(x) = \frac{3}{x}.\)
When we subtract double the second equation from the first equation, we have:
\(f(x) + 2f(\frac{1}{x})] – 2[f(\frac{1}{x}) + 2f(x)] = 3x – 2(\frac{3}{x})\)
⇔ \(f(x) + 2f(\frac{1}{x}) – 2f(\frac{1}{x}) - 4f(x) = 3x – \frac{6}{x}\)
⇔ \(-3f(x) = 3x – \frac{6}{x} \)
⇔ \(f(x) = \frac{2}{x} – x.\)
\(f(x) = f(-x)\) is equivalent to:
\(\frac{2}{x} - x = \frac{-2}{x} + x\)
⇔ \(2x = \frac{4}{x}\)
⇔ \(x^2 = 2,\)
Thus, we have \(x = ± √2.\)
Since condition 2) tells us x is an irrational number, the answer is not unique.
The conditions are not 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 not sufficient.
Therefore, E is the answer.
Answer: E
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.
12 Jun 2020, 02:18
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[GMAT math practice question]
(Function) What is the value of the function \(f(x)\)?
1) \(f(2020x + f(0)) = 2020x^2, x\) is a real number.
2) \(f(x)\) is a polynomial function.
(Function) What is the value of the function \(f(x)\)?
1) \(f(2020x + f(0)) = 2020x^2, x\) is a real number.
2) \(f(x)\) is a polynomial function.
Updated on: 21 Mar 2021, 03:01
Originally posted by MathRevolution on 14 Jun 2020, 18:51.
Last edited by MathRevolution on 21 Mar 2021, 03:01, edited 1 time in total.
Last edited by MathRevolution on 21 Mar 2021, 03:01, edited 1 time in total.
Kudos
Bookmarks
MathRevolution
=>
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.
Visit https://www.mathrevolution.com/gmat/lesson for details.
Since we have \(1\) variable (\(x\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.
Let’s look at the condition 1). It tells us that the average of \(1\) and \(3\) is \(2\). If \(x\) is between \(1\) and \(3\) inclusive, then the standard deviation of \(1, 3\), and \(x\) is less than that of \(1\) and \(2\).
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) is not sufficient, obviously.
When we consider both conditions 1) & 2) together, \(1, 2\), and \(3\) are possible values of \(x\).
The answer is not unique, and both conditions are not 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 not sufficient
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.
