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25 Feb 2019, 16:03
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
49% (02:21) correct
51%
(01:58)
wrong
based on 59
sessions
History
Date
Time
Result
Not Attempted Yet
GMATH practice exercise (Quant Class 14)
If \(\,{x^3}\left( {{x^2} + {y^2}} \right) = {z^2}\,\), is \(\,xyz = 0\,\) ?
\(\left( 1 \right)\,\,{y^3}\left( {{y^2} + {z^2}} \right) = {x^2}\,\,\)
\(\left( 2 \right)\,\,{z^3}\left( {{z^2} + {x^2}} \right) = {y^2}\)
If \(\,{x^3}\left( {{x^2} + {y^2}} \right) = {z^2}\,\), is \(\,xyz = 0\,\) ?
\(\left( 1 \right)\,\,{y^3}\left( {{y^2} + {z^2}} \right) = {x^2}\,\,\)
\(\left( 2 \right)\,\,{z^3}\left( {{z^2} + {x^2}} \right) = {y^2}\)
25 Feb 2019, 17:51
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fskilnik
\({x^3}\left( {{x^2} + {y^2}} \right) = {z^2}\,\,\,\,\,\left( * \right)\)
\(xyz\,\,\mathop = \limits^? \,\,0\)
\(\left( {1 + 2} \right)\,\,\left\{ \matrix{\\
\,{y^3}\left( {{y^2} + {z^2}} \right) = {x^2}\,\,\left( * \right) \hfill \cr \\
\,{z^3}\left( {{z^2} + {x^2}} \right) = {y^2}\,\,\left( * \right) \hfill \cr} \right.\,\,\)
\({\rm{Take}}\,\,\,\left\{ \matrix{\\
\,\left( {x;y;z} \right) = \left( {0;0;0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr \\
\,\left( {**} \right)\,\,\,\left( {x;y;z} \right) = \left( {{1 \over {\root 3 \of 2 }};{1 \over {\root 3 \of 2 }};{1 \over {\root 3 \of 2 }}} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\)
\(\left( {**} \right)\,\,{\rm{Explore}}\,\,{\rm{symmetries(!),}}\,\,\underline {{\rm{trying}}} \,\,\,\left( {x,y,z} \right) = \left( {k,k,k} \right)\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{any}}\,\,\left( * \right)} \,\,\,\,{k^3}\left( {2{k^2}} \right) = {k^2}\,\,\,\,\mathop \Rightarrow \limits^{k\, \ne \,0} \,\,\,2{k^3} = 1\,\,\,\,\, \Rightarrow \,\,\,\,k = {1 \over {\root 3 \of 2 }}\,\,\,\,{\rm{viable}}!\)
The correct answer is (E).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
23 Jan 2026, 22:49
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