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10 Jul 2018, 18:34
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[GMAT math practice question]
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
=>
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).
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
=>
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).
12 Jul 2018, 18:21
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[GMAT math practice question]
In the xy-plane, line L has equation y=mx+b, where 1<m<5. If line L passes through (1,1), which of the following points also lies on line L?
A. (2, -1)
B. (2,0)
C. (2,1)
D. (2,2)
E. (2,3)
=>
When we plug in the values x = 1 and y = 1, we obtain 1 = m + b or b = 1 – m.
All answer choices have x-coordinates of 2. Plugging 2 in for the variable x yields
y = 2m + b = 2m + 1 – m = m + 1.
Since 1 < m < 5, we must have 2 < m + 1 < 6 and 2 < y < 6.
Thus, (2,3) is the only point that can lie on the line.
Therefore, the answer is E.
Answer: E
In the xy-plane, line L has equation y=mx+b, where 1<m<5. If line L passes through (1,1), which of the following points also lies on line L?
A. (2, -1)
B. (2,0)
C. (2,1)
D. (2,2)
E. (2,3)
=>
When we plug in the values x = 1 and y = 1, we obtain 1 = m + b or b = 1 – m.
All answer choices have x-coordinates of 2. Plugging 2 in for the variable x yields
y = 2m + b = 2m + 1 – m = m + 1.
Since 1 < m < 5, we must have 2 < m + 1 < 6 and 2 < y < 6.
Thus, (2,3) is the only point that can lie on the line.
Therefore, the answer is E.
Answer: E
15 Jul 2018, 18:22
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[GMAT math practice question]
|m-n|=?
1) m and n are integers
2) mn=13
=>
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.
Since we have 2 variables (m and n) and 0 equations, C is most likely to be 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)
Since m and n are integers, the pairs of solutions of the equation mn=13 are
m = 1, n = 13; m = 13, n = 1; m = -1, n = -13; and m = -13, n = -1.
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 13, n = 1, then | m – n | = | 13 – 1 | = 12.
If m = -1, n = -13, then | m – n | = | -1 – (-13) | = 12.
If m = -13, n = -1, then | m – n | = | -13 – (-1) | = 12.
Since this question is an absolute value question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
Since we don’t have enough information, condition 1) is not sufficient.
Condition 2)
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 2, n = 13/2, then | m – n | = | 2 – 13/2 | = 9/2.
Since we don’t have a unique solution, condition 2) is not sufficient.
Therefore, C is the answer.
Answer: C
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
|m-n|=?
1) m and n are integers
2) mn=13
=>
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.
Since we have 2 variables (m and n) and 0 equations, C is most likely to be 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)
Since m and n are integers, the pairs of solutions of the equation mn=13 are
m = 1, n = 13; m = 13, n = 1; m = -1, n = -13; and m = -13, n = -1.
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 13, n = 1, then | m – n | = | 13 – 1 | = 12.
If m = -1, n = -13, then | m – n | = | -1 – (-13) | = 12.
If m = -13, n = -1, then | m – n | = | -13 – (-1) | = 12.
Since this question is an absolute value question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
Since we don’t have enough information, condition 1) is not sufficient.
Condition 2)
If m = 1, n = 13, then | m – n | = | 1 – 13 | = 12.
If m = 2, n = 13/2, then | m – n | = | 2 – 13/2 | = 9/2.
Since we don’t have a unique solution, condition 2) is not sufficient.
Therefore, C is the answer.
Answer: C
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
