11 days
A’s 1 day’s work $latex = \frac{1}{9}&s=1$
B’s 1 day’s work $latex = \frac{1}{15}&s=1$
Work done in first 2 days = A’s 1 day’s work + B’s 1 day’s work
$latex = \frac{1}{9}+\frac{1}{15} = \frac{5+3}{45} = \frac{8}{45}&s=1$
∴ Work done in first 10 days $latex = \frac{8\times 5}{45} = \frac{8}{9}&s=1$
Remaining work $latex = 1-\frac{8}{9} = \frac{1}{9}&s=1$
Now it is turn of ‘A’ for the eleventh day.
∴ Time taken by ‘A’ in doing $latex \frac{1}{9}&s=1$ work $latex = \frac{1}{9}\times 9 = 1\ day&s=1$
$latex \therefore Required\ Time\ = 10+1=11\ days$
Hence option [C] is correct answer.