For engine and refrigeration operating between two same temperature, the relationship between efficiency and coefficient of performance $\beta$,

$\eta =\frac{1}{1+\beta }$

$\frac{1}{10}=\frac{1}{1+\beta }\Rightarrow \beta =9$

From the principle of the refrigerator, coefficient of performance is given by the ratio of the amount of energy absorbed from the reservoir to work, i.e.,

$\frac{Q_2}{W}=\beta$

$\frac{Q_2}{10}=9$

$Q_2=90 \mathrm{J}$.