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Question

For r=1 roots are α,β & for r=2 roots are α,γ & for r=3 roots are β,γ, then which of the following is true?

A
α=1P1+P2P3
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B
β=1P1P2+P3
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C
γ=1P1+P2+P3
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D
1α+1β+1γ=P1+P2+P3
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Solution

The correct options are
A α=1P1+P2P3
B β=1P1P2+P3
C 1α+1β+1γ=P1+P2+P3
D γ=1P1+P2+P3
From the given data, whenr=1,2,3, we get
x22P1x+1=0 (roots, α,β)..................(A)
x22P2x+2=0(roots, α,γ)...................(B)
x32P3x+3=0(roots β,γ)...................(C)

{α+β=2P1,αβ=1,

{α+γ=4P2,αγ=2,

{β+γ=6P3,βγ=3,
For r=1,α,β are roots of x22P1x+1=0
α+β=2P1 andαβ=1
α+βαβ=2P1
1α+1β=2P1,
Similarly for r=2, we have 1α+1γ=2P2,
for r=3, we have 1β+1γ=2P3.
Add and get 2(1α+1β+1γ)=2(P1+P2+P3)
1α+1β+1γ=P1+P2+P3


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