A monoacidic base in a 0.1M solution ionizes to 0.001 %. Its ionization constant is

$K_a=\frac{C\alpha \times C\alpha }{C(1-\alpha )}$

$K_a=\frac{\left[C{H}_{3}CO{O}^{-}\right]\left[H^{+}\right]}{\left[C{H}_{3}COOH\right]}$

$K_a=\frac{C\alpha \times C\alpha }{C(1-\alpha )}$

$K_a=\frac{C{\alpha }^2}{C(1-\alpha )}$

Since $\alpha$ is very small $1-\alpha \approx$ 1

Therefore, $K_a=C{\alpha }^2$

$K_a=0.1\times (\frac{0.001}{100}{)}^2$

= $1.0\times {10}^{–11}$

Hence, the correct option is option (D) $1.0\times {10}^{–11}$.