IPMAT Indore (MCQ) PYQs

If $(1 + x - 2x^2)^6 = A_0 + \sum_{r=1}^{12} A_r x^r$, then the value of $A_2 + A_4 + A_6 + \cdots + A_{12}$ is

Let $S_1 = \{100, 105, 110, 115, ... \}$ and $S_2 = \{100, 95, 90, 85, ... \}$ be two series in arithmetic progression. If $a_k$ and $b_k$ are the $k$-th terms of $S_1$ and $S_2$, respectively, then $\sum_{k=1}^{20} a_k b_k$ equals __________.

If the sum of the first $21$ terms of the sequence: $\ln \frac{a}{b}, \ln \frac{a}{b \sqrt{b}}, \ln \frac{a}{b^{2}}, \ln \frac{a}{b^{2} \sqrt{b}}, \ldots$ is $\ln \frac{a^{m}}{b^{n}}$, then the value of $m+n$ is $\qquad$