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

Options

A. `-27`
B. `27`
C. `-54`
D. `54`

Answer

D

Explanation

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`
`c=(alpha+beta)(gamma+delta)+alpha beta +gamma delta`
`=(3+3)(3+3)+3.3+3.3=36+18=54`