Question

If α, β are the roots of the equation \[x^2 - p(x + 1) - c = 0, \text { then } (\alpha + 1)(\beta + 1) =\]

पर्याय

• c

• c − 1

•  1 − c

•  none of these

Answer 1

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 = \[\alpha\beta = - (c + p)\]

\[\text { Then }, (\alpha + 1) (\beta + 1) = \alpha\beta + \alpha + \beta + 1 \]

\[ = - (c + p) + p + 1 \]

\[ = - c - p + p + 1\]

\[ = 1 - c\]