If the roots of the equation `x^(3) + bx^(2) + cx - 1 = 0` form an increasing G.P., then b belongs to the interval

Options

A. `b+c=0`
B. `b epsilon(-oo,-3)`
C. one of the roots is 1
D. one root is smaller than one and one root is more than one

Answer

A, B, C, D

Explanation

Let the roots be `a/r,a` an `ar` where `agt0,rgt1`
`:.` Product of the roots `=1`
`impliesa/r.a.ar=1`
`impliesa^(3)=1`
`:.a=1` [ one root is 1]
Now roots are `1/4,1` and `r`. Then
`1/r+1+r=-b`
`implies1/r+r=-b-1`…..i
`:'r+1/rgt2`
`implies-b-1gt2`
`impliesblt-3`[from Eq. (i) ]
or `b epsilon(-oo,-3)`
Also `1/r.1+1.r+r . 1/r=c`
`implies1/r+r+1=c=-b` [from Eq. (i)]
`:.b+c=0`
Now first root `=1/rlt1` [ `:'` one root is smaller than one]
Second root `=1`
Third root `=rgt1` [ `:'` one root is greater tha one]