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Let omega ne 1 be a cube root of unity. Then the minimumof the set {|a+bomega+c omega^2|^2:a,b,c "distinct non -zero integers"} equals............. |
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Answer» ` =(a+bOmega +cOmega^2)bar((a+bOmega +cOmega^2)),(because zbar(z)=|z|^2)` `=(a+bOmega +cOmega^2)(a+b bar(Omega)+2c bar(Omega)^2)` `=a^2+b^2+c^2+ab)(Omega^2+Omega)+b(Omega^2+Omega^4)+ac(Omega +Omega^2)"" ["as" Omega^3=1]` ` =a^2+b^2+c^2+ab)(-1)+bc(-1)+ac(-1)"" [" as" Omega+ Omega^2=-1,Omega^4=Omega]` `=a^2+b^2+c^2-ab-bc-ca` `=(1)/(2){(a-b)^2+(b-c)^2+(c-a)^2}` `because a, b and c` are DISTINCT non - zero integers, For minimum VALUE `a=1,b=2 and c=3` `THEREFORE |a+b Omega +cOmega^2|_("min")^(2)=(1)/(2){1^2+1^2+2^2}`. ` =(6)/(2) =300`. |
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