If a and n are positive numbers, does (2a)^2x = n ?
(1) a^x + 1/a^x = sqrt (n+2)
(2) x > 0
This question is from Kaplan Prep test. Explanation is very strange and I am not able to crack it completely. Can someone help?
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Equations with three variables
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batwaraanirudh
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The question stem states: a and n are positive integers and it is asking whether (2a)^2x = n
Lets look at statement 2 first:
The information given is only about x and nothing is mentioned about a or n. With only this information we would not be able to comment anything about the relationship between a, x and n. Therefore 2 is INSUFFICIENT
Now lets have a look at statement 1:
a^x + 1/a^x = sqrt(n+2)
Lets square both the sides
=> (a^x + 1/a^x)^2 = (sqrt(n+2))^2
=>a^2x + 1/a^2x + 2 = n+2
=>a^2x + 1/a^2x = n
This is the best we can do with this statement. again we are not in a certain position to comment if 2a^2x = n+2.
If a=1, then the statement is correct else it is false.
Since we have no information about a, Therefore 1 alone is NOT SUFFICIENT
Lets look at both the statements combined:
We get information about x from this statement but not about a. so both statements together is also NOT SUFFICIENT.
Hence the answer is E
Lets look at statement 2 first:
The information given is only about x and nothing is mentioned about a or n. With only this information we would not be able to comment anything about the relationship between a, x and n. Therefore 2 is INSUFFICIENT
Now lets have a look at statement 1:
a^x + 1/a^x = sqrt(n+2)
Lets square both the sides
=> (a^x + 1/a^x)^2 = (sqrt(n+2))^2
=>a^2x + 1/a^2x + 2 = n+2
=>a^2x + 1/a^2x = n
This is the best we can do with this statement. again we are not in a certain position to comment if 2a^2x = n+2.
If a=1, then the statement is correct else it is false.
Since we have no information about a, Therefore 1 alone is NOT SUFFICIENT
Lets look at both the statements combined:
We get information about x from this statement but not about a. so both statements together is also NOT SUFFICIENT.
Hence the answer is E
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Hi Milovan,
Does the original question state that the variables a and n are "positive integers" or "positive numbers." The wording here is important.
GMAT assassins aren't born, they're made,
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Does the original question state that the variables a and n are "positive integers" or "positive numbers." The wording here is important.
GMAT assassins aren't born, they're made,
Rich
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GMATGuruNY
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I believe that the question stem should read as follows:
(x+y)² = x² + 2xy + y²
Statement 1: a^x + 1/a^x = sqrt (n+2)
a^x + a^-x = √(n+2).
Squaring both sides, we get:
(a^x)² + 2(a^x)(a^-x) + (a^-x)² = n+2
a^2x + 2(a�) + a^(-2x) = n+2
a^2x + a^(-2x) = n.
Substituting a^2x + a^(-2x) = n into the question stem -- Does 2(a^2x) = n? -- we get:
Does 2(a^2x) = a^2x + a^(-2x)?
Does a^2x = a^(-2x)?
Does a^4x = 1?
Case 1: If a=1 and x=1, the answer is YES.
Case 2: If a=2 and x=1, the answer is NO.
INSUFFICIENT.
Cases 1 and 2 also satisfy statement 2.
Thus, even when the statements are combined, the answer to the question stem can be YES or NO.
The correct answer is E.
Remember the following identity:Milovan wrote:If a and n are positive numbers, does 2(a^2x) = n ?
(1) a^x + 1/a^x = sqrt (n+2)
(2) x > 0
(x+y)² = x² + 2xy + y²
Statement 1: a^x + 1/a^x = sqrt (n+2)
a^x + a^-x = √(n+2).
Squaring both sides, we get:
(a^x)² + 2(a^x)(a^-x) + (a^-x)² = n+2
a^2x + 2(a�) + a^(-2x) = n+2
a^2x + a^(-2x) = n.
Substituting a^2x + a^(-2x) = n into the question stem -- Does 2(a^2x) = n? -- we get:
Does 2(a^2x) = a^2x + a^(-2x)?
Does a^2x = a^(-2x)?
Does a^4x = 1?
Case 1: If a=1 and x=1, the answer is YES.
Case 2: If a=2 and x=1, the answer is NO.
INSUFFICIENT.
Cases 1 and 2 also satisfy statement 2.
Thus, even when the statements are combined, the answer to the question stem can be YES or NO.
The correct answer is E.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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