$\begin{array}{l}Outofthetworootsof{x}^2+(1-2\lambda )x+({\lambda }^2-\lambda -2)=0onerootisgreaterthan3andtheotherroot\\ islessthan3,thenthe\lim itsof\lambda are\end{array}$

$\begin{array}{l}GivenEquation{x}^2+(1-2\lambda )x+({\lambda }^2-\lambda -2)=0\\ x^{2}coefficent>0andonerootgreater\\ Than3andotherlessthan3\\ \therefore f\left(3\right)<0\\ \Rightarrow 9+(1-2\lambda )3+({\lambda }^2-\lambda -2)<0\\ \Rightarrow 9+3-6\lambda +{\lambda }^2-\lambda -2<0\\ \Rightarrow {\lambda }^2-7\lambda +10<0\\ \Rightarrow (\lambda -5)(\lambda -2)<0\\ 2<\lambda <5\\ \lambda \in (2,5)\end{array}$