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Question

# If the sum of two roots of x4−2x3+4x2+6x−21=0 is zero, then which of the following is/are true?

A
one of the roots of the equation is 1+i6
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B
all roots of the equation are real
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C
the equation has only two real roots
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D
sum of all the real roots of the equation is 0
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Solution

## The correct option is D sum of all the real roots of the equation is 0 x4−2x3+4x2+6x−21=0 Let α,β,γ and δ be the roots. Then S1=α+β+γ+δ=2 S2=αβ+αγ+αδ+βγ+βδ+γδ=4 S3=αβγ+αβδ+βγδ+αγδ=−6 S4=αβγδ=−21 Let α+β=0 ⋯(1) ⇒γ+δ=2 ⋯(2) αβ+αγ+αδ+βγ+βδ+γδ=4 ⇒αβ+αγ+αδ−αγ−αδ+γδ=4 [∵β=−α] ⇒αβ+γδ=4 ⋯(3) αβγ+αβδ+βγδ+αγδ=−6 ⇒αβγ+αβδ−αγδ+αγδ=−6 [∵β=−α] ⇒αβ(γ+δ)=−6 ⇒αβ=−3 ⋯(4) αβγδ=−21 From (4),γδ=7 ⋯(5) Solving (1) and (4), we get α=√3, β=−√3 or α=−√3, β=√3 Solving (2) and (5), we get γ=1+i√6, δ=1−i√6 or γ=1−i√6, δ=1+i√6 ∴ Roots are √3,−√3,1+i√6,1−i√6

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