A) 0
B) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}\]
C) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-yz-zx-xy\]\[\]
D) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+yz+zx+xy\]
Correct Answer: C
Solution :
\[{{\omega }_{n}}=\cos \left( \frac{2\pi }{n} \right)+i\sin \left( \frac{2\pi }{n} \right)\] \[\Rightarrow {{\omega }_{3}}=\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}=-\frac{1}{2}+\frac{i\sqrt{3}}{2}=\omega \] and \[\omega _{3}^{2}={{\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)}^{2}}=\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}\] \[=-\frac{1}{2}-\frac{i\sqrt{3}}{2}={{\omega }^{2}}.\] \[\therefore \,\,\,(x+y{{\omega }_{3}}+z\omega _{3}^{2})\,(x+y\omega _{3}^{2}+z{{\omega }_{3}})\] \[=(x+y\omega +z{{\omega }^{2}})\,(x+y{{\omega }^{2}}+z\omega )\] \[={{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx\].
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