For any operation ? that acts on two numbers x and y, the commutator is defined as x?y – y?x. For which of the following operations is the commutator not equal to zero for some values of x and y?
I. x?y = xy
II. x?y = (x – y)^2
III. x?y = x^3 + 3x^2y + 3xy^2 + y^3
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III
Commutator
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I x?y = xy
so commutator = xy - yx so always 0
II x?y = (x-y)^2
so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0
III x?y = (x+y)^3
so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0
so IMO ans = C
OA Plz.
so commutator = xy - yx so always 0
II x?y = (x-y)^2
so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0
III x?y = (x+y)^3
so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0
so IMO ans = C
OA Plz.
can u pl. explain how do u get the highlighted term? I mean why it is (y-x)^3 and not (y+x)^3aspiregmat wrote:I x?y = xy
so commutator = xy - yx so always 0
II x?y = (x-y)^2
so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0
III x?y = (x+y)^3
so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0
so IMO ans = C
OA Plz.
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My mistake ...real2008 wrote:can u pl. explain how do u get the highlighted term? I mean why it is (y-x)^3 and not (y+x)^3aspiregmat wrote:I x?y = xy
so commutator = xy - yx so always 0
II x?y = (x-y)^2
so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0
III x?y = (x+y)^3
so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0
so IMO ans = C
OA Plz.
it should be (x+y)^3 - (y+x)^3 so Ans IMO none of the above , which is not in the ans choice....
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I also thought the answers should be none. This is a Manhattan Gmat challenge question I found on MBA mission. Below is the exp if anyone can make sense of it please share.
https://www.mbamission.com/blog/2009/08/ ... llenge-63/
https://www.mbamission.com/blog/2009/08/ ... llenge-63/