For a first-order reaction, the rate is directly proportional to the concentration of the reactant: $\text{Rate} = k[A]$. Therefore, the ratio of rates at two different times is equal to the ratio of concentrations at those times: $\frac{\text{Rate}_1}{\text{Rate}_2} = \frac{[A]_1}{[A]_2} = \frac{0.04}{0.03} = \frac{4}{3}$

The rate constant $k$ can be calculated using the integrated rate equation between time $t_1$ and $t_2$: $k = \frac{2.303}{t_2 - t_1} \log \frac{[A]_1}{[A]_2}$ $k = \frac{2.303}{20 - 10} \log \left(\frac{4}{3}\right) = \frac{2.303}{10} (\log 4 - \log 3)$ $k = \frac{2.303}{10} (0.6021 - 0.4771) = 0.2303 \times 0.1250 \approx 0.0287 \text{ s}^{-1}$

The half-life period ($t_{1/2}$) is given by: $t_{1/2} = \frac{0.693}{k} = \frac{0.693}{0.0287} \approx 24.14 \text{ s}$.

Note: The exact calculation yields approximately $24.1 \text{ s}$ (which was the intended option in the actual NEET exam), making $24.7 \text{ s}$ the closest available choice or a likely typographical error in the provided options.