A) \[\frac{1}{2}(-1+i\sqrt{3})\]
B) \[\frac{1}{2}(1+i\sqrt{3})\]
C) \[\frac{1}{2}(1-i\sqrt{3})\]
D) None of these
Correct Answer: A
Solution :
Since\[p<0\]. Let\[p=-q\], where \[q\]is positive. Therefore \[{{p}^{1/3}}=-{{q}^{1/3}}{{(1)}^{1/3}}.\] Hence\[\alpha =-{{q}^{1/3}}\], \[\beta =-{{q}^{1/3}}\omega \]and \[\gamma =-{{q}^{1/3}}{{\omega }^{2}}\] The given expression \[\frac{x+y\omega +z{{\omega }^{2}}}{x\omega +y{{\omega }^{2}}+z}=\frac{1}{\omega }.\frac{z\omega +y{{\omega }^{2}}+z}{x\omega +y{{\omega }^{2}}+z}\]\[=\frac{1}{\omega }={{\omega }^{2}}\].
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