$a$ and $b$ are two integers such that $10a+b=5$ and $p(x)=x^2+ax+b$.The value of n such that $p(10)p(11)=p(n)$

Question

$\begin{array}{1 1}(A)\;110&(B)\;100\\(C)\;115&(D)\;105\end{array} $

Answer 1

Let $\alpha$ and $\beta$ be the roots of $p(x)=0$

$\Rightarrow p(x)=(x-\alpha)(x-\beta)$

$p(10)p(11)=(10-\alpha)(10-\beta)(11-\alpha)(11-\beta)$

$\Rightarrow (10-\alpha)(11-\beta)(10-\beta)(11-\alpha)$

$\Rightarrow [110-10(\alpha+\beta)-\alpha+\beta\alpha][110-10(\alpha+\beta)-\beta+\beta\alpha]$

$\Rightarrow (110+10a+b-\alpha)(110+10a+b-\beta)$

$\Rightarrow (115-\alpha)(115-\beta)$

$\Rightarrow p(115)$

Hence (C) is the correct answer.