Let $ A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix} $.

If $ \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n $, $ m, n \in \mathbb{N} $, then $ m + n $ is equal to

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements

(S₁): The number of elements in R is 18, and

(S₂): The relation R is symmetric but neither reflexive nor transitive

Let f(x) be a positive function and $I_{1} = \int\limits_{-\frac{1}{2}}^{1} 2x \, f(2x(1-2x)) \, dx$ and $I_{2} = \int\limits_{-1}^{2} f(x(1-x)) \, dx$. Then the value of $\frac{I_{2}}{I_{1}}$ is equal to ________

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is