Given the function:
\[ f(x) = \frac{2x - 3}{3x - 2} \]
and if \( f_n(x) = (f \circ f \circ \ldots \circ f)(x) \) is applied \( n \) times, find \( f_{32}(x) \).
For \( n \in \mathbb{N} \), the largest positive integer that divides \( 81^n + 20n - 1 \) is \( k \). If \( S \) is the sum of all positive divisors of \( k \), then find \( S - k \).
{If \(f(x)\) is a quadratic function such that \(f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{1-x}\right)\), then \(\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} =\)}
List-I (Sentences) | List-II (Prepositions) |
(A) The edited version of her article is indistinguishable __________ her first version. | (I) With |
(B) I just don’t feel any affinity ____________ his prose style. It’s too caustic. | (II) to |
(C) Her ideas are not all that dissimilar ___________ mine. | (III) for |
(D) It would be wrong to mistake his diffidence ___________ his arrogance or coldness. | (IV) from |