A) \[{{x}^{2}}+ax-{{a}^{2}}=0\]
B) \[{{x}^{2}}+{{a}^{2}}x+a=0\]
C) \[{{x}^{2}}+ax+{{a}^{2}}=0\]
D) \[{{x}^{2}}-{{a}^{2}}x+a=0\]
Correct Answer: C
Solution :
If\[\omega \] and \[{{\omega }^{2}}\]are two imaginary cube roots of unity, then \[1+\omega +{{\omega }^{2}}=0\] \[\Rightarrow \] \[\omega +{{\omega }^{2}}=-1\] ?(i) The sum of roots \[=a{{\omega }^{317}}+a{{\omega }^{382}}\] \[=a({{\omega }^{317}}+{{\omega }^{382}})\] \[=a({{\omega }^{2}}+\omega )=-a\][from (i)] The product of roots \[=a{{\omega }^{317}}\times a{{\omega }^{382}}={{a}^{2}}{{\omega }^{699}}={{a}^{2}}\] Therefore, the required equation is \[{{x}^{2}}-\] (Sum of roots) x+ (Product of roots) = 0 \[\Rightarrow \] \[{{x}^{2}}+ax+{{a}^{2}}=0.\] Note: Cube roots of \[-1\]are \[-1,-\omega ,-{{\omega }^{2}}.\]
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