For all numbers a and b, the operation is defined by $a@b = a^2 - ab$ . If $xy \ne 0$ , then which of the following can be equal to zero?

I. $x@y$

II. $(xy)@y$

III. $x@(x + y)$

Answer Choices

Correct answer marked below

II only

I and II only

Correct

I and III only

II and III only

All of the above

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Comments

ChieduSep 23

This one looked scary at first but once I just plugged everything into the definition it wasn't bad. Key is remembering xy≠0 means neither x nor y is zero.

Praveen6d ago

Yeah the xy≠0 condition is the whole game here.

RicksonAug 10

Can someone explain why I would test each Roman numeral separately instead of just picking numbers for x and y? I feel like I'm wasting time doing it both ways.

AnketAug 11

I usually do the algebra for these, plugging in feels risky with 'can be equal to zero' wording.

ChithiraJun 3

Medium difficulty for me. II and III tripped me up because I forgot to actually expand (xy)@y before judging.

ArpitaJun 17

III is the one that got me, totally blanked on x@(x+y).

CiroSep 29

III is the interesting one since it can factor out and give you a real possibility. Nice question.

What this question tests

What it tests

Your ability to manipulate linear and quadratic expressions, substitute unknowns, and solve equations under time pressure.

Common trap

Losing a sign when moving terms across the equals sign, or dividing both sides by a variable that could equal zero.

For all numbers a and b, the operation is defined by a@b=a2−aba@b = a^2 - aba@b=a2−ab. If xy≠0xy \ne 0xy=0, then which of the following can be equal to zero?I. x@yx@yx@yII. (xy)@y(xy)@y(xy)@yIII. x@(x+y)x@(x + y)x@(x+y)