A) \[{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
B) \[{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
C) \[0\]
D) \[-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
Correct Answer: C
Solution :
\[\left| \begin{matrix} 0 & {{b}^{3}}-{{a}^{3}} & {{c}^{3}}-{{a}^{3}} \\ {{a}^{3}}-{{b}^{3}} & 0 & {{c}^{3}}-{{b}^{3}} \\ {{a}^{3}}-{{c}^{3}} & {{b}^{3}}-{{c}^{3}} & 0 \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 0 & -({{a}^{3}}-{{b}^{3}}) & -({{a}^{3}}-{{c}^{3}}) \\ {{a}^{3}}-{{b}^{3}} & 0 & -({{b}^{3}}-{{c}^{3}}) \\ {{a}^{3}}-{{c}^{3}} & {{b}^{3}}-{{c}^{3}} & 0 \\ \end{matrix} \right|\] \[=({{a}^{3}}-{{b}^{3}})[0+({{b}^{3}}-{{c}^{2}})({{a}^{3}}-{{c}^{3}})]\] \[-({{a}^{3}}-{{c}^{3}})[({{b}^{3}}-{{c}^{3}})({{a}^{3}}-{{b}^{3}})-0]\] \[=({{a}^{3}}-{{b}^{3}})({{b}^{3}}-{{c}^{3}})({{a}^{3}}-{{c}^{3}})\] \[-({{a}^{3}}-{{c}^{3}})({{b}^{3}}-{{c}^{3}})({{a}^{3}}-{{b}^{3}})\] \[=0\]
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