For all numbers a and b, let a#b=(a+b)^2-2ab. Which of the following must be true? I. x#y=y#x II. (x#y)#z=(z#x)#y III. (x+y)#(x-y)=2(x#y)

Answer Choices

Correct answer marked below

II only

III only

I and III only

Correct

II and III only

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Comments

DivyaApr 26

Thanks for the explanation. I initially missed that (a+b)^2 - 2ab simplifies to a^2 + b^2. That made the symmetry in I obvious.

RheaMay 8

Same! I was about to expand everything but the simplification saved so much time.

PeeyushJun 1

For statement II, I keep getting different results depending on the order. Can someone explain why it's not always true?

NaomiJun 6

Try plugging in simple numbers like x=1, y=2, z=3. You'll see the left and right sides don't match.

NurmuhammadJun 10

Yeah, the operation isn't associative, so moving the parentheses and swapping variables changes the value.

GowthamApr 14

This one was tricky. I had to test each statement and almost ran out of time. Definitely a 700-level question.

MeiirkhanMay 3

For III, I got 2x^2 + 2y^2 on both sides after simplifying. So III must be true. But I'm still unsure about II. Can someone confirm if II is ever true?

PusanMay 7

II is not always true. It might hold for specific numbers, but the question asks 'must be true', so it fails.

What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.