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If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0

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If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0  [#permalink]

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15 Nov 2017, 01:53
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55% (hard)

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58% (01:11) correct 42% (01:30) wrong based on 84 sessions

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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) $$f(4) = 0$$

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" PS Forum Moderator Joined: 25 Feb 2013 Posts: 1212 Location: India GPA: 3.82 If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0 [#permalink] Show Tags 15 Nov 2017, 02:08 MathRevolution wrote: [GMAT math practice question] If $$f(x) = x(x-p)(x-q)$$, is $$f(1) > 0$$? 1) $$f(2) = 0$$ 2) $$f(4) = 0$$ $$f(1)=1(1-p)(1-q)$$. this will be greater than $$0$$ only if both $$(1-p)$$ & $$(1-q)$$ are either together positive or negative. Statement 1: $$f(2)=2(2-p)(2-q)=0$$ => either $$p=2$$ or $$q=2$$. So we do not have a unique answer for $$p$$ & $$q$$. Insufficient (eg. if $$p=2$$, then $$q$$ can be positive or negative. it will not change the outcome. similarly if $$q=2$$ then $$p$$ can be anything) Statement 2: $$f(4)=4(4-p)(4-q)=0$$ => either $$p=4$$ or $$q=4$$. So we do not have a unique answer for $$p$$ & $$q$$. Insufficient combining 1 & 2 we have $$p=2$$ or $$4$$ and $$q=2$$ or $$4$$. but in either case $$(1-p)<0$$ and $$(1-q)<0$$ so we have both $$(1-p)$$ & $$(1-q)$$ as negative hence $$f(1)=1(1-p)(1-q)>0$$. Sufficient Option C Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6221 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0 [#permalink] Show Tags 17 Nov 2017, 01:11 => 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0  [#permalink]

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17 Nov 2017, 08:48
MathRevolution wrote:
=>

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).

Hi MathRevolution

Can you explain how can p & q be variable here? the function has one input x and give an output so the variable should be x. subsequently 2 & 4 should be the roots of x.
PS Forum Moderator
Joined: 25 Feb 2013
Posts: 1212
Location: India
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Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0  [#permalink]

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17 Nov 2017, 22:26
niks18 wrote:
MathRevolution wrote:
=>

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).

Hi MathRevolution

Can you explain how can p & q be variable here? the function has one input x and give an output so the variable should be x. subsequently 2 & 4 should be the roots of x.

Hi Bunuel,

it would be great if you could you provide more clarity on the concerns I have pertaining to the definition of variables for this question and solution posted above?
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6221
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0  [#permalink]

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18 Nov 2017, 16:55
niks18 wrote:
MathRevolution wrote:
=>

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).

Hi MathRevolution

Can you explain how can p & q be variable here? the function has one input x and give an output so the variable should be x. subsequently 2 & 4 should be the roots of x.

If p and q are determined, it means the function f is determined.
Since f is determined, we can figure out f(1).
That's why we have 2 variable to identify the function f(x).
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" PS Forum Moderator Joined: 25 Feb 2013 Posts: 1212 Location: India GPA: 3.82 Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0 [#permalink] Show Tags 19 Nov 2017, 06:07 Quote: Hi MathRevolution Can you explain how can p & q be variable here? the function has one input x and give an output so the variable should be x. subsequently 2 & 4 should be the roots of x. If p and q are determined, it means the function f is determined. Since f is determined, we can figure out f(1). That's why we have 2 variable to identify the function f(x). Hi MathRevolution Thanks for your reply. While i understand that to calculate the value of the function, we need the values of p & q. but should this make p & q a variable or are they simply unknown constants. The value of p & q will not change for any other input, for eg. f(3)=3(3-p)(3-q), f(5)=5(5-p)(5-q) and so on.... In my opinion this function has only one independent variable x and the dependent variable is f(x). 2 & 4 are the roots of f(x), which in turn leads to the values of p & q. this is not a multivariate function i hope. Finally even if we assume that p & q are variables, then how can we simply assume that p=2 & q=4? why can't this be other way round, I mean p=4 & q=2 ? Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6221 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0 [#permalink] Show Tags 21 Nov 2017, 10:41 niks18 wrote: Quote: Hi MathRevolution Can you explain how can p & q be variable here? the function has one input x and give an output so the variable should be x. subsequently 2 & 4 should be the roots of x. If p and q are determined, it means the function f is determined. Since f is determined, we can figure out f(1). That's why we have 2 variable to identify the function f(x). Hi MathRevolution Thanks for your reply. While i understand that to calculate the value of the function, we need the values of p & q. but should this make p & q a variable or are they simply unknown constants. The value of p & q will not change for any other input, for eg. f(3)=3(3-p)(3-q), f(5)=5(5-p)(5-q) and so on.... In my opinion this function has only one independent variable x and the dependent variable is f(x). 2 & 4 are the roots of f(x), which in turn leads to the values of p & q. this is not a multivariate function i hope. Finally even if we assume that p & q are variables, then how can we simply assume that p=2 & q=4? why can't this be other way round, I mean p=4 & q=2 ? Let's define a variable as an unknown value we need to identify, which means an unknown constant is a variable we should find. To determine the function we need to identify the values of p and q. Thus p and q are variables we need to get in order to determine the function f(x), because we can solve this question after figuring out the function f(x) _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

PS Forum Moderator
Joined: 25 Feb 2013
Posts: 1212
Location: India
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Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0  [#permalink]

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21 Nov 2017, 10:58
MathRevolution wrote:
niks18 wrote:
Quote:
Hi MathRevolution

Can you explain how can p & q be variable here? the function has one input x and give an output so the variable should be x. subsequently 2 & 4 should be the roots of x.

If p and q are determined, it means the function f is determined.
Since f is determined, we can figure out f(1).
That's why we have 2 variable to identify the function f(x).

Hi MathRevolution

Thanks for your reply. While i understand that to calculate the value of the function, we need the values of p & q. but should this make p & q a variable or are they simply unknown constants. The value of p & q will not change for any other input, for eg. f(3)=3(3-p)(3-q), f(5)=5(5-p)(5-q) and so on....

In my opinion this function has only one independent variable x and the dependent variable is f(x). 2 & 4 are the roots of f(x), which in turn leads to the values of p & q. this is not a multivariate function i hope.

Finally even if we assume that p & q are variables, then how can we simply assume that p=2 & q=4? why can't this be other way round, I mean p=4 & q=2 ?

Let's define a variable as an unknown value we need to identify, which means an unknown constant is a variable we should find.
To determine the function we need to identify the values of p and q.
Thus p and q are variables we need to get in order to determine the function f(x), because we can solve this question after figuring out the function f(x)

Hi MathRevolution,

Again thanks for your reply. Even if for the sake of argument we assume that an unknown "constant" can be termed as variable (which I am not very sure of), this does not explain the assumption that p=2 & q=4. it could be other way round as well i.e p=4 & q=2

as f(2)=0 & f(4)=0 so 2 & 4 are definitely roots of f(x) but shouldn't these be the values of x which will further lead to the values of p & q, which in this case happens to be either 2 or 4?

Thanks
Re: If f(x) = x(x-p)(x-q), is f(1) > 0? 1) f(2) = 0 2) f(4) = 0 &nbs [#permalink] 21 Nov 2017, 10:58
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