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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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If a⊙b=(a+b)^2-2ab, which of the following is(are) true?  [#permalink]

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31 Jan 2018, 02:52
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45% (medium)

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62% (01:59) correct 38% (01:41) wrong based on 89 sessions

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[GMAT math practice question]

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

_________________
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 $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Senior Manager Joined: 17 Oct 2016 Posts: 313 Location: India Concentration: Operations, Strategy GPA: 3.73 WE: Design (Real Estate) Re: If a⊙b=(a+b)^2-2ab, which of the following is(are) true? [#permalink] ### Show Tags 31 Jan 2018, 03:04 D. I and III only. a⊙b=(a+b)²−2ab =a²+b²+2ab-2ab =a²+b² Now lets test the conditions. Ⅰ.a⊙b=b⊙a a⊙b = a² + b² b⊙a = b² + a² Hence I is true Ⅱ.(a⊙b)⊙c=a⊙(b⊙c) (a⊙b)⊙c = (a² + b²)² + c² a⊙(b⊙c) = a² + (b² + c²)² clearly, (a⊙b)⊙c≠a⊙(b⊙c) II is false Ⅲ.a⊙1=a²+1² =a² + 1 III is true Hence option D _________________ Help with kudos if u found the post useful. Thanks Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8017 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If a⊙b=(a+b)^2-2ab, which of the following is(are) true? [#permalink] ### Show Tags 02 Feb 2018, 02:01 => $$a⊙b=(a+b)^2-2ab=a^2+2ab+b^2-2ab = a^2 + b^2$$ 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. Therefore, the answer is D. 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$79 for 1 month Online Course"
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Re: If a⊙b=(a+b)^2-2ab, which of the following is(are) true?   [#permalink] 02 Feb 2018, 02:01
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