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What does p equal

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Manager
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What does p equal  [#permalink]

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New post Updated on: 31 Oct 2017, 17:58
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

23% (01:31) correct 77% (01:19) wrong based on 27 sessions

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What does p equal

(1) \((\frac{p}{2}) ^ {(\frac{p}{2})} = \frac{p}{2}\)


(2) \((\frac{2}{p}) ^ {(\frac{2}{p})} = \frac{2}{p}\)

I chose D but OA is E ...

Could you confirm how would you solve this ?

thank you !

Originally posted by jabhatta@umail.iu.edu on 02 Jul 2017, 18:38.
Last edited by broall on 31 Oct 2017, 17:58, edited 4 times in total.
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Re: What does p equal  [#permalink]

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New post 02 Jul 2017, 19:22
jabhatta@umail.iu.edu wrote:
What does p equal


(1) (P/2) ^ (P/2) = (P/2)


(2) (2/P) ^ (2/P) = (2/P)

I chose D but OA is E ...

Could you confirm how would you solve this ?

thank you !



Choose case if P= 2 or P= -2.....
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Re: What does p equal  [#permalink]

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New post 02 Jul 2017, 20:05
Thanks Rohit ....

Question :

-- how long did it take you to solve this ?
-- did you try the strategy of just testing cases or did you think of some number properties theory or perhaps some theory regarding exponents

Have trouble determining if i should test cases in general or should i think of some theory to solve such problems

Thank you !
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What does p equal  [#permalink]

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New post 02 Jul 2017, 20:39
jabhatta@umail.iu.edu wrote:
What does \(p\) equal

(1) \((\frac{p}{2}) ^ {(\frac{p}{2})} = \frac{p}{2}\)


(2) \((\frac{2}{p}) ^ {(\frac{2}{p})} = \frac{2}{p}\)

I chose D but OA is E ...

Could you confirm how would you solve this ?

thank you !


To solve this question by algebra method:

Statement (1) :
To simplify the equation, set \(x=\frac{p}{2}\). We have \(x^x=x\)

Note that \(x \neq 0\) since we dont have \(0^0\)

If \(x>0\), we have
\(x\ln x=\ln x \implies (x-1)\ln x=0 \implies x=1\)

If \(x<0\), set \(t=-x \implies t>0\), we have
\((-t)^{(-t)}=-t \implies \frac{1}{(-t)^t}=-t \implies (-1)^{t+1} \times t^{t+1}=1 \implies t^{t+1}=(-1)^{t+1}\)
\(\implies t=-1 \implies x=-1\)

Hence \(x=\pm 1 \implies p=\pm 2\)

Insufficient.

Statement (2):

Solve the equation the same way as in (1), we have \(x= \pm 1 \implies p=\pm 2\). Insufficient.

Combine (1) and (2), we have \(p= \pm 2\). Insufficient.

The answer is E.

To solve quickly in GMAT, just try to find two special values of \(p\) are \(\pm 2\). We will get the answer choice E.

The trick here is that \(p \in R\) but we could consider value \(2\) only.
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What does p equal   [#permalink] 02 Jul 2017, 20:39
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