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Question

If every pair among the equations x2+ax+bc=0,x2+bx+ca=0 and x2+cx+ab=0 have a common root, then which of the following is/are correct?

A
Sum of the common roots is 12(a+b+c)
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B
Sum of the common roots is 12(a+b+c)
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C
Product of the common roots is abc
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D
Product of the common roots is abc
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Solution

The correct options are
A Sum of the common roots is 12(a+b+c)
C Product of the common roots is abc
D Product of the common roots is abc
Assuming x2+ax+bc=0(1) has roots α,β
x2+bx+ca=0(2) has roots β,γ
x2+cx+ab=0(3) has roots α,γ

Therefore,
α+β=a, αβ=bc
β+γ=b, βγ=ca
γ+α=c, γα=ab
Adding, we get
2(α+β+γ)=(a+b+c)α+β+γ=12(a+b+c)

Also, by multiplying, we get
α2β2γ2=a2b2c2αβγ=±abc

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