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805+ (Hard)|   Exponents|   Inequalities|      
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If a and n are positive numbers, does 2a^(2x) = n ? 19 Jan 2020, 23:10
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E
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45% (02:36) correct 55% (02:31) wrong based on 159 sessions

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Competition Mode Question



If a and n are positive numbers, does \(2a^{2x}=n?\)


(1) \(a^x+\frac{1}{a^x}=\sqrt{n+2}\)

(2) \(x > 0\)



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E
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If a and n are positive numbers, does 2a^(2x) = n ? Updated on: 20 Jan 2020, 23:35
Originally posted by CareerGeek on 20 Jan 2020, 04:15.
Last edited by CareerGeek on 20 Jan 2020, 23:35, edited 1 time in total.
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(1) \(a^x+\frac{1}{a^x} = \sqrt{n+2}\)
Squaring on both sides,
--> \((a^x+\frac{1}{a^x})^2 = n+2\)
--> \(a^{2x} + 2*a^x*\frac{1}{a^x} + \frac{1}{a^{2x}} = n + 2\)
--> \(a^{2x} + 2 + \frac{1}{a^{2x}} = n + 2\)
--> \(a^{2x} + \frac{1}{a^{2x}} = n\)
Does \(2a^{2x} = n\) ?
--> \(a^{2x} + \frac{1}{a^{2x}} = 2a^{2x}\)
--> \(a^{2x} = \frac{1}{a^{2x}}\)
--> \(a^{4x} = 1\)
Only Possible if \(x = 0\)
But we DO NOT know the value of x --> Insufficient

(2) \(x > 0\)
Does not talk anything about a & n --> Insufficient

Combining (1) & (2),
If \(x > 0\), \(a^{4x}\) is equal to 1 when a = 1
And not equal to 1 when a is not equal to 1
Insufficient

Option E
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 05:26
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#1
a^x+1/a^x=√n+2
possible when
a=x=1 and n=2
we get yes sufficient
#2
x>0
a and n relation not know
insufficient
IMO A
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 07:27
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Quote:
If a and n are positive numbers, does \(2a^{2x}=n\)?

(1) \(a^x+1/a^x=\sqrt{n+2}\)
(2) \(x>0\)
Show more

\(a,n>0\)

(1) \(a^x+1/a^x=\sqrt{n+2}\) insufic

\((a^x+1/a^x)^2=(\sqrt{n+2})^2…a^{2x}+a^{-2x}+2(a^x)(a^{-x})=n+2\)
\(a^{2x}+a^{-2x}+2(a^{x+(-x)})=n+2…a^{2x}+a^{-2x}+2(a^0)=n+2\)
\(a^{2x}+a^{-2x}+2=n+2…a^{2x}+a^{-2x}=n\)

\((rephrase): 2a^{2x}=n…2a^{2x}=a^{2x}+a^{-2x}\)
\(2a^{2x}-a^{2x}=a^{-2x}…a^{2x}=a^{-2x}\)
\(a^{2x}=1/a^{2x}…a^{2x}a^{2x}=1…a^{4x}=1\)

\((rephrase):a^{4x}=1?…yes:a=1…or…x=0\)

(2) \(x>0\) insufic

(1&2) insufic

\(x>0…x≠0\) but is \(a=1\)?
\(a=1,x>0:a^{4x}=1…1^{anything}=1…answer=yes\)
\(a=2,x>0:a^{4x}=1…2^{m4}≠1…answer=no\)

Ans (E)
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 07:30
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Is \(2a^{2x} = n\)?

1) Squaring both sides, one will obtain:
\(a^{2x} +a^{-2x}+2=n+2\)
\(a^{2x} +a^{-2x}=n\)

If x=0, then 1+1=2=n, then it must be true that \(2a^{2x}= 2 = n\).

If x>0, then \(2a^{2x} =2n-2a^{-2x} \). Thus, it is not clear that \(2a^{2x} = n\). For clarity, consider a=1 or a=2.

NOT SUFFICIENT

2) We have no useful information on a and n.
NOT SUFFICIENT

1)+2) Since x>0, \(2a^{2x} =2n-2a^{-2x} \). Thus, it could be true (or false) that \(2a^{2x} = n\).
NOT SUFFICIENT

FINAL ANSWER IS (E)
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 08:05
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If a and n are positive numbers, does 2a^2x=n?


(1) a^x+1/(a^x)=root(n+2)
a^x+a^-x=root(n+2)
(a^x+a^-x)^2=n+2
a^2x+2a^x a^-x+a-2x=n+2
a^2x+1+a^-2x=n+2
a^(2x)+a^(-2x)-1=n
they not equal
sufficient


(2) x>0
we don't have any clue about a and n
insufficient

Therefore, A
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 10:35
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If a and n are positive numbers, does \(2a^{2x} = n\)?

(1) \(a^x+\frac{1}{a^x} = √(n+2)\)
\(a^{2x}+\frac{1}{a^{2x}} +2 = n+2\)
\(a^{2x}+\frac{1}{a^{2x}} = n\)
a, x and n can take any value so

INSUFFICIENT.

(2) x > 0
None of the values are given.

INSUFFICIENT.

Together 1 and 2
INSUFFICIENT.

Answer E.
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 18:34
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1) from this we won't be getting 2a^2x =n....... sufficient to say that they are not equal....but since they are positive no's...for a=1.....n becomes 2......for both of them....so insufficient

2) clearly insufficient

OA:E
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If a and n are positive numbers, does 2a^(2x) = n ? 20 Jan 2020, 19:44
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Ans: E

a)a^x + (1/a^x) = sqrt (n+2)
squaring both the sides
a^2x+(1/a^2x)+2=n+2
a^2x+(1/a^2x)=n
if x=0
n=2 and, 2a^(2x) = n
x=0, so, 2a^0=2...true
if x=1,a=1,n=2 and equation holds..true
if x=1,a=2,n=17/4 and equation doesn't hold..false
not sufficient

b)x>0....alone clearly not sufficient

combined..for,x=1,a=1..true
x=1,a=2..false

So, taking both not sufficient.
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If a and n are positive numbers, does 2a^(2x) = n ? 27 Jul 2022, 05:08
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