A carnot’s engine operating between temperatures $T_{1}and{T}_2$ has efficiency $\frac{1}{6}$ . When $T_2$  is lowered by 62K , its efficiency increases to $\frac{1}{3}$ . Then $T_{1}and{T}_2$ are respectively

$case\left(i\right)$   $\eta =1-\frac{T_2}{T_1}$

                                $\frac{1}{6}=1-\frac{T_2}{T_1}-\left(1\right)$

                $case\left(ii\right)$    ${\eta }^1=1-\frac{T_2^1}{T_1}$

                                        ${\eta }^1=1-\frac{\left(T_2-62\right)}{T_1}$

                                        $=1-\frac{T_2}{T_1}+\frac{62}{T_1}$

                                        ${\eta }^1=\eta +\frac{62}{T_1}$

                By given date  ${\eta }^1=2n$

                                        $2\eta =\eta +\frac{62}{T_1}$

                $n=\frac{62}{T_1}\Rightarrow T_1=372k$

                $\Rightarrow \left(1\right)\Rightarrow \frac{1}{6}=1-\frac{T_2}{372}$

                        $T_2=310k$

                $T_1=372k\&T_2=310k.$