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Question

If roots of cubic equation are in G.P. , ax3+bx2+cx+d then :

A
c3a=b3d
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B
a2c=b2d
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C
ac2=bd2
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D
NOTA
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Solution

The correct option is A c3a=b3d
LetIfα,β,andγarethethreerootsthenwemusthaveax3+bx2+cx+d=a(xα)(xβ)(xγ)Comparingcoefficientsofvariouspowerofxonbothsidesleadstoα+β+γ=ba(1)αβ+βγ+γα=ca(2)αβγ=da(3)Onsubstitutingβ=αrandγ=αr2wegetα(1+r+r2)=ba(4)α2(r+r2+r3)=ca(5)α3r3=da(6)Now,dividing(5)by(4)toαr=cbandsubstitutingthisin(6)weget(cb)3=dac3b3=dac3a=b3d.
Hence, the option A is the correct answer.

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