If α, β are the zeros of the polynomial f(x) = x² – p (x + 1) – c such that (α + 1) (β + 1) = 0, then c =
A. 1
B. 0
C. – 1
D. 2
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C) -1
x² - p ( x + 1 ) + c
⇒ x² - p x - p + c
⇒ x² - p x + ( c - p )
Comparing with ax² + bx + c, we get :
a = 1
b = - p
c = c - p .
Given :
( α + 1 )( β + 1 ) = 0
⇒ αβ + α + β + 1 = 0
Note that, sum of roots = - b/a
α + β = - b / a
But b = - p
a = 1
So α + β = - ( - p ) / 1 = p
Product of roots = αβ = c / a
⇒ αβ = ( c - p )
Hence write this as :
αβ + α + β + 1 = 0
⇒ c - p + p + 1 = 0
⇒ c + 1 = 0
⇒ c = -1
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