A) \[\frac{a}{b}\]
B) b
C) \[\omega \]
D) \[{{\omega }^{2}}\]
Correct Answer: C
Solution :
1, \[\omega \] and \[{{\omega }^{2}}\]are the three cube roots of unity. \[\Rightarrow \,\,1+\omega +{{\omega }^{2}}=0\,\,and\,\,{{\omega }^{3}}=1\] The given expression \[\frac{a{{\omega }^{6}}+b{{\omega }^{4}}+c{{\omega }^{2}}}{b+c{{\omega }^{10}}+a{{\omega }^{8}}}=\frac{a+b\omega +c{{\omega }^{2}}}{b+c\omega +a{{\omega }^{2}}}\] \[[{{\omega }^{6}}=1,\,\,{{\omega }^{4}}=\omega ]\] \[=\,\,\frac{\omega (a+b\omega +c{{\omega }^{2}})}{\omega (b+c\omega +a{{\omega }^{2}})}\,\] \[[Multiplying\,\,{{N}^{r}}\,\,and\,\,{{D}^{r}}\,\,by\,\,\omega ]\] \[=\,\,\frac{\omega (a+b\omega +c{{\omega }^{2}})}{(a{{\omega }^{3}}+b\omega +c{{\omega }^{2}})}=\frac{\omega (a+b\omega +c{{\omega }^{2}})}{(a+b\omega +c{{\omega }^{2}})}=\omega \]
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