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If roots of the equations x^(4)-12x^(3)+...

If roots of the equations `x^(4)-12x^(3)+cx^(2)+dx+81=0` are positive.
The value of d is

A

`-1`

B

`-1/2`

C

`1`

D

`1/2`

Text Solution

Verified by Experts

The correct Answer is:
C
Let roots be `alpha, beta, gamma, delta gt 0`. ltrgt `:.alpha +beta+gamma+delta=12` ltbr. `(alpha+beta)(gamma+delta)+alpha beta+gamma delta=c`
`alpha beta(gamma+delta)+gamma delta(alpha+beta)=-d`
`alpha beta gamma delta =81`
`:'AM=(alpha+beta+gamma+delta)/4=3`
and `GM=(alpha beta gamma delta)^(1//4)=(81)^(1//4)=(81)^(1//4)=3`
`:.AM=GM`
`impliesalpha =beta= gamma = delta =3`
Required root `=-d/(2c)=-((-108))/(2xx54)=-1`
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