Question

If $ \omega $ and $ {{\omega }^{2}} $ are the two imaginary cube root unity, then the equation whose roots are $ a{{\omega }^{317}} $ and $ a{{\omega }^{382}} $ is:

• 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 $

Answer 1

Answer: (C)

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}}. $