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Question

Let α and β be the roots of the equation x2+x+1=0. The equation whose roots are α19,β7 is

A
x2x1=0
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B
x2x+1=0
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C
x2+x1=0
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D
x2+x+1=0
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Solution

The correct option is B x2+x+1=0
x2+x+1=0

(xω)(xω2)=0

x=ω,ω2
α=ω,β=ω2 (ω,ω2 are cube roots of unity )
Hence, α3=ω3=1
β3=[ω3]2=1
αβ=ω3=1
α19=(α3)6α=16α=α=ω and β7=β6.β=12.β=β=ω2
α19+β7=ω+ω2=1
α19β7=ω.ω2=ω3=1
Hence equation whose roots are α19,β7 is
x2(α19+β7)x+α19β7=0

x2+x+1

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