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Question

If α,β,γ be the non zero real roots of the equation x3+px2+qx+r=0 satisfying the relation αβ+1=0, then

A
γ=r
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B
r2+pr+q=1
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C
1α+1β=p+r
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D
Harmonic mean of α,β,γ=3rq
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Solution

The correct options are
A γ=r
B 1α+1β=p+r
C Harmonic mean of α,β,γ=3rq
D r2+pr+q=1
α,β,γ be the non zero real roots of the equation x3+px2+qx+r=0
Then, α+β+γ=p
and αβγ=r
γ=r(αβ+1=0)
Also, αβ+(α+β)γ=q
1+(pγ)γ=q
1(p+r)r=q
r2+pr+q=1
1α+1β=α+βαβ=pr1=p+r
H.M of α,β,γ=31α+1β+1γ
=3p+r+1r=3rpr+r2+1
=3rq=3rq.

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