According to the formula, If person P can do a work in x days and person Q can do the same work in y days then, one day work of P and Q together is $\frac{\mathrm{xy}}{x+y}$

so, $\frac{10\times 12}{10+12}=\frac{120}{22}=5\frac{5}{11}\mathrm{days}$

Hence, is both of them work together, work will be completed in $5\frac{5}{11}$days

Hence option A is correct