A) \[\frac{1}{\sqrt{7}}\]
B) \[\frac{\sqrt{7}}{8}\]
C) \[\frac{\sqrt{7}}{2}\]
D) None of these
Correct Answer: C
Solution :
[c] \[1,\,{{\omega }_{1}},\,{{\omega }_{2}}.....,{{\omega }_{6}}\] are 7th roots of unity in the increasing order of argument. Let \[{{\omega }_{1}}+{{\omega }_{2}}+{{\omega }_{4}}=\alpha +i\beta \] ?...(1) then \[{{\omega }_{6}}+{{\omega }_{5}}+{{\omega }_{3}}=\alpha -i\beta \] ?...(2) (as \[{{\omega }_{1}}\] and \[{{\omega }_{6}},{{\omega }_{2}}\]and \[{{\omega }_{5}},{{\omega }_{4}}\] and \[{{\omega }_{3}}\]are conjugate pairs) \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\alpha =-1\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha =-\frac{1}{2}\] Multiplying (1) and (2), we get \[{{\alpha }^{2}}+{{\beta }^{2}}={{\omega }_{7}}+{{\omega }_{6}}+{{\omega }_{4}}+{{\omega }_{8}}+{{\omega }_{7}}+{{\omega }_{5}}+{{\omega }_{10}}\] \[+{{\omega }_{9}}+{{\omega }_{7}}\] \[=3+({{\omega }_{6}}+{{\omega }_{4}}+{{\omega }_{8}}+{{\omega }_{5}}+{{\omega }_{10}}+{{\omega }_{9}})\] \[=3+({{\omega }_{6}}+{{\omega }_{4}}+{{\omega }_{1}}+{{\omega }_{5}}+{{\omega }_{3}}+{{\omega }_{2}})\] \[=3+(-1)\] \[=2\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\beta =\frac{\sqrt{7}}{2}\]
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