$\begin{array}{l}Thenumberofallpossiblepositive\mathrm{int}egralvaluesof\alpha forwhichtherootsofthequadraticequation,\\ 6x^2-11x+\alpha =0arerationalnumbersis:\end{array}$

$\begin{array}{l}Givenrootsof6x^2-11x+\alpha =0arerational\\ \Rightarrow Dmustbeperfectsquare\\ \Rightarrow D=121-24\alpha \\ If\alpha =3,thenD=121-72=49isperfectsquare\\ If\alpha =4,thenD=121-96=25isperfectsquare\\ If\alpha =5,thenD=121-120=1isperfectsquare\\ \therefore 3\mathrm{int}egralvalues.\end{array}$