We have,

Mean $=\frac{x_1+x_2+x_3+x_4+x_5}{5}=10$

$\Rightarrow x_1+x_2+x_3+x_4+x_5=50 ...\left(\mathrm{i}\right)$

Variance

$={\left(\mathrm{S}.\mathrm{D}\right)}^2=\frac{\mathrm{\Sigma }{x}_i^2}{5}-{\left(\overline{x}\right)}^2=9$ 

$\Rightarrow \frac{x_1^2+x_2^2+x_3^2+x_4^2+x_5^2}{5}-100=9$

$\Rightarrow x_1^2+x_2^2+x_3^2+x_4^2+x_5^2=545 ...\left(\mathrm{ii}\right)$

Now,

${\overline{x}}_{\mathrm{n}\mathrm{e}\mathrm{w}}=\frac{x_1+x_2+x_3+x_4+x_5-50}{6}=0$ 

New variance

$=\frac{{\displaystyle \sum _{i=1}^6}{x}_i^2}{6}-{\left({\overline{x}}_{\mathrm{n}\mathrm{e}\mathrm{w}}\right)}^2$

$=\frac{x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+2500}{6}-0$

$=\frac{545+2500}{6}=\frac{3045}{6}=507.5$