Official solution
Given that,
Mean of a, b, 8, 5, 10 = 6
\[\therefore\,\,\,\] \[{{a + b + 8 + 5 + 10} \over 5} = 6\]
\[ \Rightarrow \,\,\,\] a + b + 23 = 30
\[ \Rightarrow \,\,\,\] a + b = 7 ....... (1)
Variance \[ = {{\sum {{{\left( {{x_i} - A} \right)}^2}} } \over n} = 6.8\]
\[ \Rightarrow \,\,\,{{{{\left( {6 - a} \right)}^2} + {{\left( {6 - b} \right)}^2} + {{\left( {6 - 8} \right)}^2} + {{\left( {6 - 5} \right)}^2} + {{\left( {6 - 10} \right)}^2}} \over 5} = 6.5\]
\[ \Rightarrow \,\,\,{\left( {6 - a} \right)^2} + {\left( {6 - b} \right)^2} + 4 + 1 + 16 = 34\]
\[ \Rightarrow {\left( {6 - a} \right)^2} + {\left( {6 - b} \right)^2} = 13\]
\[ \Rightarrow \,\,\,{a^2} + {b^2} = 25\]
By using (1) we get,
\[{a^2} + \left( {7 - a} \right){}^2 = 25\]
\[ \Rightarrow \,\,\,a{}^2 - 7a + 12 = 0\]
\[\therefore\,\,\,\] a = 3, 4
then b = 4, 3.