A) only \[\frac{1}{a}+\frac{1}{b\omega }+\frac{1}{c{{\omega }^{2}}}=0\]
B) only \[\frac{1}{a}+\frac{1}{b{{\omega }^{2}}}+\frac{1}{c\omega }=0\]
C) \[\frac{1}{a\omega }+\frac{1}{b{{\omega }^{2}}}+\frac{1}{c}=0\]
D) All of the above
Correct Answer: D
Solution :
We have, \[\left| \begin{matrix} bc & ca & ab \\ ca & ab & bc \\ ab & bc & ca \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[{{(ab)}^{3}}+{{(bc)}^{3}}+{{(ca)}^{3}}-3(ab)(bc)(ca)=0\] \[\Rightarrow \]\[(ab+b{{\omega }^{2}}.c+ca\omega )(ab\omega +bc{{\omega }^{2}}+ca)\] \[(ab{{\omega }^{2}}+bc\omega +ca)=0\] \[\Rightarrow \]\[ab+bc{{\omega }^{2}}+ca\omega =0,\]\[ab\omega +bc{{\omega }^{2}}+ca=0,\] \[ab{{\omega }^{2}}+bc\omega +ca=0\] \[\Rightarrow \]\[\frac{1}{c{{\omega }^{2}}}+\frac{1}{a}+\frac{1}{b\omega }=0,\] \[\frac{1}{c\omega }+\frac{1}{a}+\frac{1}{b{{\omega }^{2}}}=0,\]\[\frac{1}{c}+\frac{1}{a\omega }+\frac{1}{b{{\omega }^{2}}}=0\] \[\Rightarrow \]\[\frac{1}{a}+\frac{1}{b\omega }+\frac{1}{c{{\omega }^{2}}}=0,\frac{1}{a}+\frac{1}{b{{\omega }^{2}}}+\frac{1}{c\omega }=0,\] \[\frac{1}{a\omega }+\frac{1}{b{{\omega }^{2}}}+\frac{1}{c}=0\]
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