`x^(3)+5x^(2)+px+q=0` and `x^(3)+7x^(2)+px+r=0` have two roos in common. If their third roots are `gamma_(1)` and `gamma_(2)` , respectively, then `|g

Question

`x^(3)+5x^(2)+px+q=0` and `x^(3)+7x^(2)+px+r=0` have two roos in common. If their third roots are `gamma_(1)` and `gamma_(2)` , respectively, then `|g

`x^(3)+5x^(2)+px+q=0` and `x^(3)+7x^(2)+px+r=0` have two roos in common. If their third roots are `gamma_(1)` and `gamma_(2)` , respectively, then `|gamma_(1)-gamma_(2)|=`
A. `10`
B. `12`
C. `13`
D. `42`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - B
`(b)` Let the roots of `x^(3)+5x^(2)+px+q=0` are `alpha_(1)`, `beta_(1)`, `gamma_(1)`……..`(i)`
Then roots of the `x^(3)+7x^(2)+px+r=0` is `alpha_(1)`, `beta_(1)`, `gamma_(2)`……..`(ii)`
From `(i)`-`(ii)`, `-2x^(2)+q-r=0`
This equation has roots `alpha_(1)`, `beta_(1)`
`implies alpha_(1)+beta_(1)=0`
Now from `(i) alpha_(1)+beta_(1)+gamma_(1)=-5impliesgamma_(1)=-5`
From `(ii) alpha_(1)+beta_(1)+gamma_(2)=-7impliesgamma_(2)=-7`
`:. gamma_(1)+gamma_(2)=-12`
`:. |gamma_(1)+gamma_(2)|=12`