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Question

If α,β,γ are the roots of x3x21=0, then 
  1. 1+α1α+1+β1β+1+γ1γ=5
  2. 11α+11β+11γ=1
  3. αβ+βγ+γα=0
  4. α+β+γ=0


Solution

The correct options are
A 1+α1α+1+β1β+1+γ1γ=5
B 11α+11β+11γ=1
C αβ+βγ+γα=0
x3x21=0
From the given equation,
α+β+γ=1αβ+βγ+γα=0αβγ=1

(xα)(xβ)(xγ)=x3x21
Taking log on both sides,
ln(xα)+ln(xβ)+ln(xγ)=ln(x3x21)
Differentiating w.r.t. x and putting x=1
1xα+1xβ+1xγ=3x22xx3x2111α+11β+11γ=1


1+α1α+1+β1β+1+γ1γ=2(1α)1α+2(1β)1β+2(1γ)1γ=2(11α+11β+11γ)3=5

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