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

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If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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19 Jan 2020, 22:10
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41% (02:36) correct 59% (02:12) wrong based on 44 sessions

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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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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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Updated on: 20 Jan 2020, 22:35
1
(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

Originally posted by Dillesh4096 on 20 Jan 2020, 03:15.
Last edited by Dillesh4096 on 20 Jan 2020, 22:35, edited 1 time in total.
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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 04:26
#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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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 06:27
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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) $$x>0$$

$$a,n>0$$

(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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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 06:30
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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

Kindly give kudos if you like the explanation
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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 07:05
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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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 09:35
1
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.

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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 17:34
1
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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Re: If a and n are positive numbers, does 2a^(2x) = n ?  [#permalink]

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20 Jan 2020, 18:44
1
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.
Re: If a and n are positive numbers, does 2a^(2x) = n ?   [#permalink] 20 Jan 2020, 18:44
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