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M28-37

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Bunuel
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M28-37 [#permalink] New post   16 Sep 2014, 01:31
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A
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E

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75% (hard)

Question Stats:

33% (01:08) correct 67% (01:04) wrong based on 193 sessions
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If \(x\) is not equal to 0 and \(x^y=1\), then which of the following must be true?

I. \(x=1\)

II. \(x=1\) and \(y=0\)

III. \(x=1\) or \(y=0\)


A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)
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E
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Bunuel
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Re M28-37 [#permalink] New post   16 Sep 2014, 01:31
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Official Solution:


If \(x\) is not equal to 0 and \(x^y=1\), then which of the following must be true?

I. \(x=1\)

II. \(x=1\) and \(y=0\)

III. \(x=1\) or \(y=0\)


A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)


\(x^y=1\) implies we have one of the following three cases:

1. The base is 1, because \(1^y=1\), for all \(y\);

2. The exponent is 0, because \(x^0=1\), for any nonzero \(x\);

3. The base is -1 and the exponent is even, because \((-1)^x=1\), for any even \(x\)

Notice that if we have case 3, so if \(x=-1\) and \(y\) is any even number, then \((-1)^{even}=1\), and in this case none of the options must be true.


Answer: E
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KarishmaB
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M28-37 [#permalink] New post   Updated on: 08 Aug 2023, 04:20
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AbhiR3 wrote:
KarishmaB @scotttargesttestprep can you help us here. How do we deal with MUST be and COULD be true questions.

Posted from my mobile device


Yes, I have discussed this in detail in this post:
https://anaprep.com/algebra-must-be-tru ... questions/

Focus clearly on 'what is given' and 'what is asked'.

Originally posted by KarishmaB on 17 Mar 2022, 01:47.
Last edited by KarishmaB on 08 Aug 2023, 04:20, edited 1 time in total.
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Re: M28-37 [#permalink] New post   17 Mar 2022, 02:02
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Bunuel wrote:
If \(x\) is not equal to 0 and \(x^y=1\), then which of the following must be true?
I. \(x=1\)

II. \(x=1\) and \(y=0\)

III. \(x=1\) or \(y=0\)


A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(None\)


What is given? x is not 0 and x^y = 1

What is asked?

I. \(x=1\)

Is it necessary that x is equal to 1? No. x could be 2 and y could be 0 and still x^y = 1.


II. \(x=1\) and \(y=0\)

Is it necessary that x is equal to 1 and y is equal to 0? No. As seen above,

III. \(x=1\) or \(y=0\)

Is it necessary that x is equal to 1 or y is equal to 0? (Is it necessary that at least one of them is true?)
No. If x = -1 and y = 2, still x^y = 1

Hence, none of the statements needs to be necessarily true. Answer (E)
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coreyander
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Re: M28-37 [#permalink] New post   09 Jul 2020, 11:25
I think this is a high-quality question and I agree with explanation. An excellent question
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Re: M28-37 [#permalink] New post   31 May 2021, 05:18
Nice Qus

Posted from my mobile device
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kri1311
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Re: M28-37 [#permalink] New post   03 Jul 2021, 11:01
This question looks a bit ambiguous to me. I had selected B because if x =1 and y = 0, x^y will always be 1.
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Bunuel
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Re: M28-37 [#permalink] New post   03 Jul 2021, 11:16
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kri1311 wrote:
This question looks a bit ambiguous to me. I had selected B because if x =1 and y = 0, x^y will always be 1.


No ambiguity there. The question asks "which of the following must be true?". x^y = 1 does not necessarily means that x = 1 and y = 0, there are infinitely many other possibilities, so B is not always true.
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Re: M28-37 [#permalink] New post   17 Mar 2022, 01:27
KarishmaB @scotttargesttestprep can you help us here. How do we deal with MUST be and COULD be true questions.

Posted from my mobile device
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Bunuel
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Re: M28-37 [#permalink] New post   03 Mar 2023, 12:33
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M28-37 [#permalink]
03 Mar 2023, 12:33
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