Home
Class 12
MATHS
If the roots of the equation x^(3) + bx^...

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

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

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
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]
Promotional Banner

Similar Questions

Explore conceptually related problems

If the roots of the equation x^(3) + ax^(2) + bx + c = 0 are in A.P., 2a^(3) - 9ab =

If p and q are the roots of the equation x^(2)+p x+q=0 then

The roots of the quadratic equations ax^(2)+ bx =0 are:

If the roots of the cubic equation x^(3) -9x^(2) +a=0 are in A.P., Then find one of roots and a

If p and q are the roots of the equation x^2-p x+q=0 , then

The sum of the roots of the quadratic equations ax^(2) + bx + c=0 is

The nature of the roots of the equations ax^(2) + bx+ c =0 is decided by:

Find the sum of roots of quadratic equations ax^(2) +bx+ c=0

If (1 + 2i) is a root of the equation x^(2) + bx + c = 0 . Where b and c are real, then (b,c) is given by :