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

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

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New post 31 Jan 2018, 01:52
1
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

59% (01:55) correct 41% (01:41) wrong based on 84 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

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

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New post 31 Jan 2018, 02:04
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
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Math Revolution GMAT Instructor
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Re: If a⊙b=(a+b)^2-2ab, which of the following is(are) true?  [#permalink]

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New post 02 Feb 2018, 01:01
=>

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

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
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Re: If a⊙b=(a+b)^2-2ab, which of the following is(are) true? &nbs [#permalink] 02 Feb 2018, 01:01
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If a⊙b=(a+b)^2-2ab, which of the following is(are) true?

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