Question

If α, β are the roots of the equation $x^2-p(x+1)-c=0,\mathrm{then}(\alpha +1)(\beta +1)=$
(a) c
(b) c − 1
(c) 1 − c
(d) none of these

Answer 1

(c) 1 − c
Given equation: $$\begin{gathered} x^2-p(x+1)-c=0 \\ \mathrm{or}{x}^2-px-p-c=0 \end{gathered}$$
Also $\alpha and\beta$ are the roots of the equation.
Sum of the roots = $\alpha +\beta =p$
Product of the roots = $\alpha \beta =-(c+p)$


$$\begin{gathered} \mathrm{Then,}(\alpha +1)(\beta +1)=\alpha \beta +\alpha +\beta +1 \\ =-(c+p)+p+1 \\ =-c-p+p+1 \\ =1-c \end{gathered}$$