Let $\mathrm{A}=\left[\begin{array}{ccc}1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1\end{array}\right]$ be a singular matrix. Let $f(x)=\int_0^x\left(\mathrm{t}^2+2 \mathrm{t}+3\right) \mathrm{dt}, x \in[1, \alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1, \alpha]$, then $3(\mathrm{M}-\mathrm{m})$ is equal to :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be such that $f(x y)=f(x) f(y)$, for all $x, y \in \mathbf{R}$ and $f(0) \neq 0$. Let $g:[1, \infty) \rightarrow \mathbf{R}$ be a differentiable function such that

$$ x^2 g(x)=\int_1^x\left(\mathrm{t}^2 f(\mathrm{t})-\operatorname{tg}(\mathrm{t})\right) d t $$

Then $g(2)$ is equal to :

The area of the region $\left\{(x, y): x^2-8 x \leq y \leq-x\right\}$ is :

The value of the integral $\int\limits_{-1}^1\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right) \mathrm{d} x$ is equal to: