A) \[1\]
B) \[2\]
C) \[3\]
D) \[4\]
Correct Answer: A
Solution :
Given equation is \[{{x}^{3}}-{{x}^{2}}+x-1=0\] \[\Rightarrow \] \[{{x}^{3}}-1-{{x}^{2}}+x=0\] \[\Rightarrow \] \[(x-1)({{x}^{2}}+x+1)-x(x-1)=0\] \[\Rightarrow \] \[(x-1)({{x}^{2}}+1)=0\] \[\Rightarrow \] \[x=1,\,\,i,\,-i\] Let \[\alpha =1,\,\,\beta =i\] and \[\gamma =-i\] Then, \[{{\alpha }^{-3}}+{{\beta }^{-3}}+{{\gamma }^{-3}}={{(1)}^{-3}}+{{(i)}^{-3}}+{{(-i)}^{-3}}\] \[=1+\frac{1}{{{i}^{3}}}-\frac{1}{{{i}^{3}}}=1\]
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