$\text{ A function }f\left(x\right)\text{ defines as }f\left(x\right)=\left\{\begin{array}{ll}{x}^2+ax+1& x\text{ is rational }\\ a{x}^2+2x+b& x\text{ is irrational }\end{array}\text{ is continuous at }x=1\text{ and }2\right.$$\text{ then }(a,b)=$

$x=1\text{ and }x=2\text{ are roots of the equation }$

$\begin{array}{l}{x}^2+ax+1=a{x}^2+2x+b\\ \text{ Or }(a-1)x^2+(2-a)x+b-1=0\\ \text{ Consider the sum of the roots and product of the roots }\frac{a-2}{a-1}=3\text{ and }\frac{b-1}{a-1}=2 \Rightarrow a=\frac{1}{2}\text{ and }\mathrm{b}=0\end{array}$