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Is x > sqrt(3)? (1) 3^(x) = sqrt(27) (2)x^3 + x^5 + x^7

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Is x > sqrt(3)? (1) 3^(x) = sqrt(27) (2)x^3 + x^5 + x^7 [#permalink] New post 07 Mar 2011, 08:44
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Q. Is x > sqrt(3)?

(1) 3^(x) = sqrt(27)

(2) x^3 + x^5 + x^7 = 3591/128


The point of the second statement is to acknowledge that no matter how great one is at math, there will always be something one can't handle. How can this question be confidently answered?

[Reveal] Spoiler:
The strategy here is to recognize what needs to be done to answer the question. The questions can be sufficiently answered in two way, 1) yes, x is greater than the square root of three, or 2) no, x is not greater than the square root of three.

Never does the exact value of x come into play. One might be tempted to solve the first statement for x. But, only a human calculator could attempt to solve the second statement.

But one needn't be a math person to know that a single equation containing a single variable that is not being absolute valued or raised to an even exponent will have a unique solution on the GMAT.

So, applied here, this means Statement (1) is SUFFICIENT and Statement (2) is SUFFICIENT, since each will yield one value for x. It is irrelevant whether x is greater than the square root of three. The answer is EACH statement ALONE is sufficient.

The point here is twofold. First, one shouldn't do unnecessary work to answer a question that is solvable by simply understanding the imposed restrictions of the GMAT. Second, everyone gets stumped by a math problem sometimes. But if one knows lateral ways of attacking questions, then any mathematical blind spots are less damaging.
[Reveal] Spoiler: OA

Last edited by spacelandprep on 07 Mar 2011, 10:14, edited 1 time in total.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 09:28
spacelandprep wrote:
Q. Is x > sqrt(3)?

(1) 3^(-x) = sqrt(27)

(2) log (3^x) - cos (x*pi) = x^(-1/3)

NOTE: Logarithms and trigonometry are not concepts tested on the GMAT.

NOTE: Data sufficiency statements will not contradict one another as they do in this example.

The point of the second statement is to acknowledge that no matter how great one is at math, there will always be something one can't handle. How can this question be confidently answered?

[Reveal] Spoiler:
The strategy here is to recognize what needs to be done to answer the question. The questions can be sufficiently answered in two way, 1) yes, x is greater than the square root of three, or 2) no, x is not greater than the square root of three.

Never does the exact value of x come into play. One might be tempted to solve the first statement for x, which deals with concepts tested on the GMAT. But, only a human calculator could attempt to solve the second statement.

But one needn't be a math person to know that a single equation containing a single variable that is not being absolute valued or raised to an even exponent will have a unique solution on the GMAT.

So, applied here, this means Statement (1) is SUFFICIENT and Statement (2) is SUFFICIENT, since each will yield one value for x. It is irrelevant whether x is greater than the square root of three. The answer is EACH statement ALONE is sufficient.

The point here is twofold. First, one shouldn't do unnecessary work to answer a question that is solvable by simply understanding the imposed restrictions of the GMAT. Second, everyone gets stumped by a math problem sometimes. But if one knows lateral ways of attacking questions, then any mathematical blind spots are less damaging.


Clearly, statement 1 gives a unique solution for x, so sufficient to determine whether x is > sqrt(3)

On statement 2, while it appears to be one equation in one variable, it is not really the case as logx and cosx are both functions that can be expressed in terms of a polynomial of x. For e.g. cosx = 1-x^2/2!+x^4/4! -....and so on, so cant say for sure that there would be only one unique solution for the equation in stem 2. For all we know there can be solutions which lie on both sides of sqrt(3), so cannot be determined unless we can solve the equation and I don't think GMAT expects this equation to be solved.

It is not a good example for the point being made about not needing to actually solve the problem.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 09:34
Brilliant ! yes cannot hear a better explanation. But just as in CR you cannot touch the evidence here, you cannot touch the statements. They are assumed to be true and don't contradict each other. First statement tells me clearly that x is negative. In any case to prove two values of x with one exponent - the exponent should be raised to an even power.

e.g x^3 = -1. Proves that x = -1. Since i dont see any even powers here, I may conclude from 2) alone that x has one value. But again this question is not upto the gmat standards.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 09:52
gmat1220 wrote:
Brilliant ! yes cannot hear a better explanation. But just as in CR you cannot touch the evidence here, you cannot touch the statements. They are assumed to be true and don't contradict each other. First statement tells me clearly that x is negative. In any case to prove two values of x with one exponent - the exponent should be raised to an even power.

e.g x^3 = -1. Proves that x = -1. Since i dont see any even powers here, I may conclude from 2) alone that x has one value. But again this question is not upto the gmat standards.


We cannot be certain of the highlighted portion above. As i said, cosx actually expands into even powers of x, so more than one roots are certainly possible. Further, if you just put this expression in the spread sheet, you would see that at least three values of x (one very close to -2, one very close to 1 and one very close to 3.387) would make it valid.. there may be many others.

So, 2nd stem is fairly ambiguous
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 10:16
Great points, guys. There's really no reason for the problem to be outside the realm of the GMAT. It has been changed to address your thoughts.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 12:51
spacelandprep wrote:
But one needn't be a math person to know that a single equation containing a single variable that is not being absolute valued or raised to an even exponent will have a unique solution on the GMAT.


That's not true at all. Even a simple equation like

x^3 = x

has three solutions for x, -1, 0 and 1. When you have an equation in which x is raised to various positive integer powers, you can have as many solutions as the highest power in the equation. It doesn't make any difference whether the powers are odd or even.

There are other situations besides the ones you describe where you can have multiple solutions. For example, the equation

1^x = 1

has an infinite number of solutions. And while it's beyond the scope of the GMAT, it's especially true of equations involving trigonometric functions (like the one in the example you first provided, then deleted) that you can have multiple solutions; the equation cos(x) = 0, for example, has an infinite number of solutions.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 16:01
Excellent inputs from all. Thanks Ian and beyondgmatscore
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 21:12
On the GMAT, equations involving single-variables without even exponents or absolute values virtually always* solve to a single value. Knowing this saves time and can bail out a GMAT taker when the equation is beyond the know-how she or he possesses.

It is important to recognize that the GMAT is not a math test--it's a problem solving test. Limiting oneself to mathematical knowledge is limiting one's potential.

*An rare example from the Official Guide in which a single-variable-not-raised-to-an-even-exponent-or-absolute-valued equation has multiple solutions is PS #215, but even it has only one solution on the GMAT.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 07 Mar 2011, 22:37
[quote="spacelandprep"]Q. Is x > sqrt(3)?

(1) 3^(x) = sqrt(27)

(2) x^3 + x^5 + x^7 = 3591/128


Statement 1: x=3/2=1.5 which is less than sqrt(3).Hence statement is sufficient.

Statement 2:x^3(1+x^2+x^4)=3591/128

3591/128 can be expressed as 3^3*7*19/2^7
comparing LHS and RHS x=3/2 which is less than sqrt(3).Hence 2 is also sufficient.

So Ans is D
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 08 Mar 2011, 12:03
spacelandprep wrote:
An rare example from the Official Guide in which a single-variable-not-raised-to-an-even-exponent-or-absolute-valued equation has multiple solutions is PS #215, but even it has only one solution on the GMAT.


If such examples are rare, it's because single-variable equations almost never appear in GMAT Data Sufficiency algebra questions. You can count on the fingers of two hands the number of such equations that appear in the DS section of the Official Guide and the Official Quant Review combined, which makes it debatable whether there's even any value to learning 'tricks' for guessing in such situations. And despite the minuscule number of such equations in the official guides, looking at Q30 in the DS section of the OG, Statement 1 has no absolute values or even powers, and yet gives two values for n, and Statement 2 does have an even power, yet gives only one value for n. If you were to apply the heuristic 'if there are no absolute values or even exponents, there's one solution', you're going to get this question wrong. It's early in the book as well, so it's not a difficult question.

The GMAT question designers are well aware of the overly simplistic 'tricks' that many prep companies encourage test takers to use, and they design questions to trap people who apply these tricks without understanding the underlying mathematics. For example, many books suggest counting equations and unknowns, and say that you need to have at least as many equations as unknowns to solve. Q123 (the infamous 'stamps question') in the DS section of the OG is one of dozens of examples I could give of official questions designed to trap the test taker who simply counts equations and unknowns.

spacelandprep wrote:
It is important to recognize that the GMAT is not a math test--it's a problem solving test. Limiting oneself to mathematical knowledge is limiting one's potential.


What the GMAT certainly is *not* is a test of how many prep company tricks you can learn. It would defeat the entire purpose of the test if you could do well at it by learning a few 'tricks' from a prep book. It *is*, at least in part, a test of mathematical ability - not of computational ability, but rather of the ability to reason logically about mathematical concepts. You simply cannot do well on the Quant section of the GMAT without learning some math.
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Re: Spaceland Prep Strategy Question #2 [#permalink] New post 08 Mar 2011, 15:25
IanStewart wrote:
You simply cannot do well on the Quant section of the GMAT without learning some math.


Absolutely. Learn the content. Learn the strategies. Succeed on the GMAT.
Re: Spaceland Prep Strategy Question #2   [#permalink] 08 Mar 2011, 15:25
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