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Question

If all the real numbers x1,x2,x3, satisfying the equation x3x2+βx+γ=0 are in A.P.

Then,all possible values of β belong to

A
(,13)
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B
(,13)
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C
(13,)
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D
(13,)
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Solution

The correct option is A (,13)
Comparing given equation with the general form of a cubic equation (ax3+bx2+cx+d)=0,

we have
a=1,b=1,c=β,d=γ

Let the roots of the equation be a1d,a1 and a1+d.

Sum of the roots=ba=(1)1=1=(a1d)+(a1)+(a1+d)=3a1a1=13..........(i)

Sum of bi-product of roots

ca=β1=β=(a1d)(a1)+(a1d)(a1+d)+(a1)(a1+d)=a12a1d+a12d2+a12+a1d=3a12d2

=3(13)2d2=13d2..........from (i)

β=13d2 d2=13β

Since square of a number is always non-negative,we can imply (13β)0 β13β(,13)

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