A) \[\frac{1-i\sqrt{3}}{2}\]
B) \[\frac{-1-i\sqrt{3}}{2}\]
C) \[(x+y+z)i\]
D) \[pi\]
E) \[\frac{x+y+z}{2}pi\]
Correct Answer: B
Solution :
As\[p<0,\]therefore\[p=-q,\]where\[q>0\] \[\therefore \]\[{{p}^{1/3}}={{(-q)}^{1/3}}={{q}^{1/3}}{{(-1)}^{1/3}}\] \[\Rightarrow \]\[{{p}^{1/3}}=-{{q}^{1/3}},-{{q}^{1/3}}\omega ,-{{q}^{1/3}}{{\omega }^{2}}\] \[\therefore \]\[\frac{x\alpha +y\beta +z\gamma }{x\beta +y\gamma +z\alpha }=\frac{x+y\omega +z{{\omega }^{2}}}{x\omega +y{{\omega }^{2}}+z}={{\omega }^{2}}\] \[=\frac{-1-i\sqrt{3}}{2}\]
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