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If `alpha` and `beta` are the roots of the equation `x^2-p(x+1)-q=0` then the value of `(alpha^2+2alpha+1)/(alpha^2+2alpha+q)` + `(beta^2+2beta+1)/(beta^2+2beta+q)` is (A)1 (B) 2 (C) 3 (D) 0

Text Solution

Verified by Experts

The correct Answer is:
A
`alpha+beta=p` and `alphabeta=-(p+q)`
Now, `(alpha + 1) (beta+1) =(alpha +beta)+alphabeta+1=1-q`
The given expression =`((alpha+1)^2)/((alpha+1)^2+(q-1))+((beta+1)^2)/((beta+1)^2+(q-1))`
`=((alpha+1)^2)/((alpha+1)^2 -(alpha +1) (beta+1))+((beta+1)^2)/((beta+1)^2-(alpha +1)(beta+1))`
`=((alpha +1))/((alpha -beta))+((beta+1))/((beta-alpha ))=((alpha -beta))/((alpha -beta))=1`
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