Let $1+\sum _{\mathrm{r}=1}^{10} \left(3^{\mathrm{r}}{\cdot }^{10}{\mathrm{C}}_{\mathrm{r}}+\mathrm{r}{\cdot }^{10}{\mathrm{C}}_{\mathrm{r}}\right)=2^{10}\left(\mathrm{\alpha }\cdot 4^5+\mathrm{\beta }\right)$ where $\mathrm{\alpha }, \mathrm{\beta }\in \mathrm{N}$ and $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2-2\mathrm{x}-{\mathrm{k}}^2+1\text{ if }\mathrm{\alpha }, \mathrm{\beta }$ lies between the roots of $\mathrm{f}\left(\mathrm{x}\right)=0$, then the smallest positive integral value of k is

$\begin{array}{l}1+\sum _{\mathrm{r}=1}^{10} \left(3^{\mathrm{r}}{\cdot }^{10}{\mathrm{C}}_{\mathrm{r}}+\mathrm{r}{\cdot }^{10}{\mathrm{C}}_{\mathrm{r}}\right)=2^{10}\left(\mathrm{\alpha }\cdot 4^5+\mathrm{\beta }\right)\\ \Rightarrow 4^{10}+10\sum _{\mathrm{r}=1}^{10}{ }^9{\mathrm{C}}_{\mathrm{r}-1}\Rightarrow 4^{10}+10{.2}^9\\ =2^{10}\left[\mathrm{\alpha }{4}^5+\mathrm{\beta }\right]=2^{20}+2^{10}\left[\mathrm{\alpha }{4}^5+\mathrm{\beta }\right]\\ \Rightarrow 2^{10}\left[4^5+5\right]=2^{10}\left[\mathrm{\alpha }{4}^5+\mathrm{\beta }\right]\Rightarrow \mathrm{\alpha }=1; \mathrm{\beta }=5.\\ \Rightarrow \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2-2\mathrm{x}-{\mathrm{k}}^2+1\\ {\mathrm{x}}_1\mathrm{\alpha }\mathrm{\beta }{\mathrm{x}}_2\\ \mathrm{f}\left(1\right)<0\mathrm{f}\left(5\right)<025-10-{\mathrm{k}}^2+1<0\\ 1-2-{\mathrm{k}}^2+<0 16-{\mathrm{k}}^2<0\\ \mathrm{k}>0\mathrm{K}=5\end{array}$