Solution
\[
\text { old Mean }=\frac{\sum \mathrm{x}_{1}}{10}=10 \Rightarrow \sum \mathrm{x}_{1}=100
\]
\[
\begin{aligned}
& \text { New Mean }=\frac{\sum x_{i}-\alpha+\beta}{10}=\frac{100-\alpha+\beta}{10}=10.1 \\
& \Rightarrow 100-\alpha+\beta=101 \Rightarrow \beta=1+\alpha \\
& \text { Old Variance }=\frac{1}{n} \sum x_{i}^{2}-(\text { Mean })^{2}=2 \\
& \Rightarrow \frac{1}{10} \sum x_{i}^{2}-100=2 \Rightarrow \sum x_{i}^{2}=1020 \\
& \text { New Variance }=\frac{1}{10}\left(\sum x_{i}^{2}-\alpha^{2}+\beta^{2}\right)-\binom{\text { New }}{\text { Mean }}^{2}=1.99 \\
& \Rightarrow \frac{1020-\alpha^{2}+\beta^{2}}{10}-(10.1)^{2}=1.99 \\
& \Rightarrow \frac{1020-\alpha^{2}+\beta^{2}}{10}=\frac{101 \times 101}{100}+\frac{199}{100} \\
& \Rightarrow 1020-\alpha^{2}+\beta^{2}=\frac{1}{10}\left(100^{2}+1+200+199\right) \\
& \Rightarrow 1020-\alpha^{2}+(1+\alpha)^{2}=1040 \\
& \Rightarrow \alpha=\frac{19}{2}, \beta=1+\alpha=\frac{21}{2} \\
& \alpha+\beta=\frac{19+21}{2}=20 \text { Answer(4) }
\end{aligned}
\]
.
