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:

Let $\mathrm{R}=\left\{(x, y) \in \mathbf{N} \times \mathbf{N}: \log _{\mathrm{e}}(x+y) \leq 2\right\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is $\_\_\_\_$ .

If $\left(1-x^3\right)^{10}=\sum\limits_{\mathrm{r}=0}^{10} \mathrm{a}_{\mathrm{r}} x^{\mathrm{r}}(1-x)^{30-2 \mathrm{r}}$, then $\frac{9 \mathrm{a}_9}{\mathrm{a}_{10}}$ is equal to $\_\_\_\_$ .