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M61-16

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M61-16  [#permalink]

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New post 18 Jun 2018, 03:37
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A
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E

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Question Stats:

67% (01:45) correct 33% (00:00) wrong based on 3 sessions

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If \(a⊙b=(a+b)^2-2ab\), which of the following is (are) true?



Ⅰ. \(a⊙b=b⊙a\)

Ⅱ. \((a⊙b)⊙c=a⊙(b⊙c)\)

Ⅲ. \(a⊙1=a^2+1\)


A. Ⅰ only
B. Ⅱ only
C. Ⅲ only
D. Ⅰ and Ⅲ only
E. Ⅰ, Ⅱ and Ⅲ

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Re M61-16  [#permalink]

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New post 18 Jun 2018, 03:37
Official Solution:


If \(a⊙b=(a+b)^2-2ab\), which of the following is (are) true?



Ⅰ. \(a⊙b=b⊙a\)

Ⅱ. \((a⊙b)⊙c=a⊙(b⊙c)\)

Ⅲ. \(a⊙1=a^2+1\)


A. Ⅰ only
B. Ⅱ only
C. Ⅲ only
D. Ⅰ and Ⅲ only
E. Ⅰ, Ⅱ and Ⅲ


\(a⊙b=(a+b)^2-2ab=a^2+2ab+b^2-2ab = a^2 + b^2\)

Statement I

\(b⊙a = b^2+a^2 = a^2 + b^2 = a⊙b\)

Therefore, statement I is true.

Statement II

\((a⊙b)⊙c = (a^2+b^2) ⊙c = (a^2+b^2) ^2 +c^2 = a^4 + 2a^2b^2 + b^4 + c^2\)

\(a⊙(b⊙c) = a⊙ (b^2+c^2) = a^2+ (b^2+c^2) ^2 = a^2 + b^4 + 2b^2c^2 + c^4\)

We can easily find a counterexample. If \(a = 1\), \(b = 2\) and \(c = 3\), then \((a⊙b)⊙c = (1⊙2)⊙3 = (1^2+2^2) ⊙3 = 5⊙3 = 5^2 + 3^2 = 25 + 9 = 34\) and \(a⊙(b⊙c) = 1⊙(2⊙3) = 1⊙(2^2+3^2) = 1⊙13 = 1^2+13^2 = 1 + 169 = 170\).

Thus, statement II is false.

Statement III

\(a⊙1 = a^2 + 1^2 = a^2 + 1\)

Therefore, statement III is true.


Answer: D
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Re: M61-16  [#permalink]

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New post 15 Sep 2018, 05:48
Can someone explain how Statement 3 is true ?
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Re: M61-16 &nbs [#permalink] 15 Sep 2018, 05:48
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