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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7372
GMAT 1: 760 Q51 V42 GPA: 3.82
M61-16  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 67% (01:49) correct 33% (00:51) wrong based on 6 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

_________________
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7372
GMAT 1: 760 Q51 V42 GPA: 3.82
Re M61-16  [#permalink]

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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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Joined: 01 Apr 2018
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GPA: 3.66
Re: M61-16  [#permalink]

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Can someone explain how Statement 3 is true ? Re: M61-16   [#permalink] 15 Sep 2018, 06:48
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# M61-16

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