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Question

Let α,β be the roots of x2+x+1=0. If 1a+α+1b+α+1c+α=2β and 1a+β+1b+β+1c+β=2α, then 1a+1+1b+1+1c+1 is equal to

A
2α2β
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B
2αβ2
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C
αβ2
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D
2αβ
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Solution

The correct options are
A 2α2β
B 2αβ2
D 2αβ
Roots of x2+x+1=0 are ω,ω2 where ω is the cube root of unity.
Let α=ω, β=ω2
Now,
1a+α+1b+α+1c+α=2β
1a+ω+1b+ω+1c+ω=2ω [ω3=1]

1a+β+1b+β+1c+β=2α1a+ω2+1b+ω2+1c+ω2=2ω2

So, the equation
1a+x+1b+x+1c+x=2x (1)
has roots ω,ω2

Simplifying the equation, we get
x[(x+a)(x+c)+(x+a)(x+b)+(x+b)(x+c)] =2(x+a)(x+b)(x+c)x[3x2+2(a+b+c)x+]=2[x3+(a+b+c)x2+]x3+0x2+=0
The roots of the cubic equations are ω,ω2,γ, so

Sum of roots
ω+ω2+γ=0γ=1 (ω+ω2=1)

So, x=1 is also a root of equation (1), putting it in equation (1), we get
1a+1+1b+1+1c+1=2
Now,
αβ=ω3=1α2β=1αβ2=1

Hence,
1a+1+1b+1+1c+1=2αβ=2α2β=2αβ2

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