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Question

Equation x4+ax3+bx2+cx+1=0 has real roots (a,b,c are non-negative). Minimum non-negative real value of b is

A
12
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B
15
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C
6
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D
10
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Solution

The correct option is D 6
Given that a polynomial x4+ax3+bx2+cx+1=0 has real roots.

Let α1,α2,α3,α4 be the roots of the polynomial.

Therefore, α1α2α3α4=1
α1+α2+α3+α4=a
α1α2+α1α3+α1α4+α2α3+α2α4+α3α4=b
α1α2α3+α1α3α4+α1α2α4+α2α3α4=c

We know that A.M.G.M.

therefore, α1α2+α1α3+α1α4+α2α4+α2α3+α3α466α31α32α33α34

b66(α1α2α3α4)3

b66(1)3

b61

b6

Therefore the minimum value of b is 6

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