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Which of the following is(are) true for ab=ba?

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Which of the following is(are) true for ab=ba?  [#permalink]

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New post 24 Feb 2016, 22:16
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A
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Question Stats:

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Which of the following is(are) true for a@b=b@a?

I. a@b=ab+ba
II. a@b=(a+b)(a-b)
III. a@b=(a/b)+(b/a)

A. only Ⅰ
B. only Ⅱ
C. only Ⅲ
D. only Ⅰ& Ⅲ
E. only Ⅱ & Ⅲ

* A solution will be posted in two days.

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Re: Which of the following is(are) true for ab=ba?  [#permalink]

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New post 25 Feb 2016, 00:49
1
MathRevolution wrote:
Which of the following is(are) true for a@b=b@a?

I. a@b=ab+ba
II. a@b=(a+b)(a-b)
III. a@b=(a/b)+(b/a)

A. only Ⅰ
B. only Ⅱ
C. only Ⅲ
D. only Ⅰ& Ⅲ
E. only Ⅱ & Ⅲ



I. a@b=ab+ba
a@b = ab + ba
b@a = ba + ab
Hence a@b = b@a

II. a@b=(a+b)(a-b)
a@b = (a+b)(a-b) = a^2 - b^2
b@a = (b+a)(b-a) = b^2 - a^2
Hence a@b ≠ b@a

III. a@b=(a/b)+(b/a)
a@b=(a/b)+(b/a) = (a^2 + b^2)/ab
b@a = b/a + a/b = (a^2 + b^2)/ab
Hence a@b = b@a

Therefore only I and III hold true.
Option D
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Re: Which of the following is(are) true for ab=ba?  [#permalink]

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New post 27 Feb 2016, 20:23
Which of the following is(are) true for a@b=b@a?

I. a@b=ab+ba
II. a@b=(a+b)(a-b)
III. a@b=(a/b)+(b/a)

A. only Ⅰ
B. only Ⅱ
C. only Ⅲ
D. only Ⅰ& Ⅲ
E. only Ⅱ & Ⅲ


-> If I. a@b=ab+ba, a@b=ab+ba=ba+ab=b@a (O).
If II. a@b=(a+b)(a-b), a@b=(a+b)(a-b)≠(b+a)/(b-a)=b@a (X).
If III.a@b=(a/b)+(b/a), a@b=(a/b)+(b/a)=(b/a)+(a/b)=b@a (O).
Thus, Ⅰ& Ⅲ are the answers. Therefore, D is the answer.
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Re: Which of the following is(are) true for ab=ba?  [#permalink]

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New post 12 Jan 2019, 22:23
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Re: Which of the following is(are) true for ab=ba?   [#permalink] 12 Jan 2019, 22:23
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