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

Options

A. `(-3, oo)`
B. `(-oo, -3)`
C. `(-1, oo)`
D. `(-oo, -1)`

Answer

B

Explanation

Let a/r, a, ar be the roots of the given equation, where r `gt` 1. Then,
`(a)/(r)+a+ar = -b" "...(i)`
and, `(a)/(r) xx a xx ar = 1 rArr a = 1`
Putting a = 1 in (i), we get
`therefore" "(1)/(r) + 1 + r = -b" "...(ii)`
`rArr" "r^(2) + r(b+1) + 1 = 0" "[because r in R]`
`rArr" "(b+1)^(2) - 4 gt 0`
`rArr" "(b+3)(b-1) gt 0`
`rArr" "b lt - 3 or, b gt 1 rArr b in (-oo, -3)" "[because b lt 0 "from (ii)"]`