Difference between revisions of "2006 AMC 12A Problems/Problem 2"

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(Removed bogus answer. h=1 is a bad choice of substitution since you can't distinguish between h, or h^3.)
 
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<math>\textbf{(A)}\ -h\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ h\qquad\textbf{(D)}\ 2h\qquad\textbf{(E)}\ h^3</math>
 
<math>\textbf{(A)}\ -h\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ h\qquad\textbf{(D)}\ 2h\qquad\textbf{(E)}\ h^3</math>
  
== Solution ==
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== Solution 1 ==
By the definition of <math>\otimes</math>, we have <math>h\otimes h=h^{3}-h</math>. Then <math>h\otimes (h\otimes h)=h\otimes (h^{3}-h)=h^{3}-(h^{3}-h)=h</math>. The answer is <math>\mathrm{(C)}</math>.
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By the definition of <math>\otimes</math>, we have <math>h\otimes h=h^{3}-h</math>.  
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Then, <math>h\otimes (h\otimes h)=h\otimes (h^{3}-h)=h^{3}-(h^{3}-h)=\boxed{\textbf{(C) }h}.</math>
  
==Solution 2==
 
Substitute <math>1</math> for <math>h</math>. You get <math>1^3-(1^3-1)</math> which is <math>1</math>. That is <math>h</math>, so the answer is <math>\mathrm{(C)}</math>.
 
~dragoon also aop is a god
 
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2006|ab=A|num-b=1|num-a=3}}
 
{{AMC12 box|year=2006|ab=A|num-b=1|num-a=3}}

Latest revision as of 13:03, 27 July 2023

The following problem is from both the 2006 AMC 12A #2 and 2006 AMC 10A #2, so both problems redirect to this page.

Problem

Define $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$?

$\textbf{(A)}\ -h\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ h\qquad\textbf{(D)}\ 2h\qquad\textbf{(E)}\ h^3$

Solution 1

By the definition of $\otimes$, we have $h\otimes h=h^{3}-h$.

Then, $h\otimes (h\otimes h)=h\otimes (h^{3}-h)=h^{3}-(h^{3}-h)=\boxed{\textbf{(C) }h}.$

See also

2006 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions
2006 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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