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Question
If α, β are the roots of the equation \[x^2 - p(x + 1) - c = 0, \text { then } (\alpha + 1)(\beta + 1) =\]
Options
c
c − 1
1 − c
none of these
Solution
1 − c
Given equation:
\[x^2 - p(x + 1) - c = 0 \]
\[or x^2 - px - p - c = 0\]
Also
\[\alpha \text { and } \beta\] are the roots of the equation.
Sum of the roots = \[\alpha + \beta = p\]
Product of the roots = \[\alpha\beta = - (c + p)\]
\[\text { Then }, (\alpha + 1) (\beta + 1) = \alpha\beta + \alpha + \beta + 1 \]
\[ = - (c + p) + p + 1 \]
\[ = - c - p + p + 1\]
\[ = 1 - c\]
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