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# M61-16

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7372
GMAT 1: 760 Q51 V42
GPA: 3.82

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18 Jun 2018, 04:37
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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

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7372 GMAT 1: 760 Q51 V42 GPA: 3.82 Re M61-16 [#permalink] ### Show Tags 18 Jun 2018, 04:37 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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Joined: 01 Apr 2018
Posts: 8
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15 Sep 2018, 06:48
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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