“Between the year 1946 and the year 1955, I did not file any income tax returns.” With that [1] statement, Ramesh embarked on an account of his encounter with the income tax department. "I originally owed Rs. 20,000 in unpaid taxes. With [2] and [3], the 20,000 became 60,000. The Income Tax Department then went into action, and I learned first hand just how much power the Tax Department wields. Royalties and trust funds can be [4]; automobiles may be [5], and auctioned off. Nothing belongs to the [6] until the case is settled.”
Match List-I with List-II and choose the correct option:
\[ \begin{array}{|l|l|} \hline \textbf{LIST-I (Function)} & \textbf{LIST-II (Expansion)} \\ \hline A. \log(1-x) & I. 1 + \frac{1}{3} + \frac{1}{6} + \frac{3}{40} + \frac{15}{336} + \dots \\ \hline B. \sin^{-1} x & II. 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \\ \hline C. \log 2 & III. x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} + \dots, -1 < x \le 1 \\ \hline D. \frac{\pi}{2} & IV. -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots, -1 \le x < 1 \\ \hline \end{array} \]
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: