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15 Nov 2017, 01:50
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[GMAT math practice question]
In the x-y plane y=p(x)=ax2+bx+c pass through (-1,0) and (1,0). Is p(0)<0?
1) a>0
2) b2-4ac>0
=>
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution
p(-1) = a – b + c = 0
p(1) = a + b + c = 0
Since we have 3 variables, a, b and c and 2 equations, D is most like to be the answer.
Condition 1)
11.13.png [ 7.33 KiB | Viewed 4845 times ]
Since the leading coefficient a is positive, the parabola is concave up and p(0) < 0.
This is sufficient.
Condition 2)
It means the graph intersects x-axis twice and that the discriminant is positive.
We have two cases that the parabola could be concave up or concave down.
For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.
Answer: A
In the x-y plane y=p(x)=ax2+bx+c pass through (-1,0) and (1,0). Is p(0)<0?
1) a>0
2) b2-4ac>0
=>
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution
p(-1) = a – b + c = 0
p(1) = a + b + c = 0
Since we have 3 variables, a, b and c and 2 equations, D is most like to be the answer.
Condition 1)
Attachment:
11.13.png [ 7.33 KiB | Viewed 4845 times ]
Since the leading coefficient a is positive, the parabola is concave up and p(0) < 0.
This is sufficient.
Condition 2)
It means the graph intersects x-axis twice and that the discriminant is positive.
We have two cases that the parabola could be concave up or concave down.
For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.
Answer: A
16 Nov 2017, 01:17
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[GMAT math practice question]
Is x < 0?
1) x^5+x^3+1<0
2) x^3+1 < 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.
Since the question includes 1 variable (x) and 0 equations, D is most likely to be the answer.
Condition 1)
x^5+x^3+1<0 implies that x^3(x^2+1)< -1.
Since x^2+1>0 always, x^3< -1/x^2+1<0.
Therefore, x<0.
This is sufficient.
Condition 2)
x^3+1 < 0 is equivalent to (x+1)(x^2-x+1)<0. Since
x^2-x+1>0 always, it follows that x+1<0.
This is sufficient, too.
Therefore, the answer is D, as expected.
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.
Answer: D
Is x < 0?
1) x^5+x^3+1<0
2) x^3+1 < 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.
Since the question includes 1 variable (x) and 0 equations, D is most likely to be the answer.
Condition 1)
x^5+x^3+1<0 implies that x^3(x^2+1)< -1.
Since x^2+1>0 always, x^3< -1/x^2+1<0.
Therefore, x<0.
This is sufficient.
Condition 2)
x^3+1 < 0 is equivalent to (x+1)(x^2-x+1)<0. Since
x^2-x+1>0 always, it follows that x+1<0.
This is sufficient, too.
Therefore, the answer is D, as expected.
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.
Answer: D
17 Nov 2017, 01:11
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[GMAT math practice question]
If f(x) = x(x-p)(x-q), is f(1) > 0?
1) f(2) = 0
2) f(4) = 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.
Since the question includes 2 variables (p and q) and 0 equations, C is most likely to be the answer.
Conditions 1) & 2):
The equations f(2) = 0 and f(4) = 0 tell us that 2 and 4 are roots of x(x-p)(x-q) = 0, and so f(x) = x(x-2)(x-4)
Therefore, f(1) = 1(1-2)(1-4) = 1*(-1)*(-3) = 3 > 0, and the answer is ‘yes’.
Therefore, conditions 1) and 2), when taken together, are sufficient.
The answer is C, as expected.
Normally, in problems which require 2 or more additional 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).
Answer: C
If f(x) = x(x-p)(x-q), is f(1) > 0?
1) f(2) = 0
2) f(4) = 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.
Since the question includes 2 variables (p and q) and 0 equations, C is most likely to be the answer.
Conditions 1) & 2):
The equations f(2) = 0 and f(4) = 0 tell us that 2 and 4 are roots of x(x-p)(x-q) = 0, and so f(x) = x(x-2)(x-4)
Therefore, f(1) = 1(1-2)(1-4) = 1*(-1)*(-3) = 3 > 0, and the answer is ‘yes’.
Therefore, conditions 1) and 2), when taken together, are sufficient.
The answer is C, as expected.
Normally, in problems which require 2 or more additional 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).
Answer: C
