A) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
B) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
C) \[(a+b+c)({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]
D) \[(a+b+c)(ab+bc+ca)\]
E) \[abc\]
Correct Answer: B
Solution :
\[(b+c)(bc)\cos A+(a+c)(ac)\cos B\] \[+(a+b)(ab)\cos C\] \[=(b+c)\left( \frac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc} \right)bc+(a+c)(ac)\] \[\times \left( \frac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ca} \right)+(a+b)(ab)\left( \frac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab} \right)\] \[=\frac{1}{2}\{(b+c)({{b}^{2}}+{{c}^{2}}-{{a}^{2}})+(a+c)\] \[\times ({{a}^{2}}+{{c}^{2}}-{{b}^{2}})+(a+b)({{a}^{2}}+{{b}^{2}}-{{c}^{2}})\}\] \[=\frac{1}{2}\{{{b}^{3}}+b{{c}^{2}}-{{a}^{2}}b+{{b}^{2}}c+{{c}^{3}}-{{a}^{2}}c\] \[+{{a}^{3}}+{{c}^{2}}a-{{b}^{2}}a+c{{a}^{2}}+{{c}^{3}}-c{{b}^{2}}+{{a}^{3}}+a{{b}^{2}}\] \[-a{{c}^{2}}+{{a}^{2}}b+{{b}^{3}}-b{{c}^{2}})\] \[=\frac{1}{2}\{2{{a}^{3}}+2{{b}^{3}}+2{{c}^{3}}\}={{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
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