Let $$\mathrm{A}=\{1,2,3,4,5\}$$. Let $$\mathrm{R}$$ be a relation on $$\mathrm{A}$$ defined by $$x \mathrm{R} y$$ if and only if $$4 x \leq 5 \mathrm{y}$$. Let $$\mathrm{m}$$ be the number of elements in $$\mathrm{R}$$ and $$\mathrm{n}$$ be the minimum number of elements from $$\mathrm{A} \times \mathrm{A}$$ that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

Let $$f(x)=\frac{1}{7-\sin 5 x}$$ be a function defined on $$\mathbf{R}$$. Then the range of the function $$f(x)$$ is equal to :

$$\lim _\limits{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}$$ is equal to :

If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}+2 \mathrm{n}$$ is equal to :