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

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 $\qquad$

The $3^{\text {rd }}, 14^{\text {th }}$ and $69^{\text {th }}$ terms of an arithmetic progression form three distinct and consecutive terms of a geometric progression. If the next term of the geometric progression is the $n^{\text {th }}$ term of the arithmetic progression, then $n$ equals ________.