The area bounded by the $$\mathrm{X}$$-axis and the curve $$y=x(x-2)(x+1)$$ is

Let $$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$$ and $$\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$$ be continuous functions. Then the value of the integral $$\int_\limits{\frac{-\pi}{2}}^{\frac{\pi}{2}}[\mathrm{f}(x)+\mathrm{f}(-x)][\mathrm{g}(x)-\mathrm{g}(-x)] \mathrm{d} x$$ is

If $$\bar{p}=\hat{i}+\hat{j}+\hat{k}$$ and $$\bar{q}=\hat{i}-2 \hat{j}+\hat{k}$$. Then a vector of magnitude $$5 \sqrt{3}$$ units perpendicular to the vector $$\bar{q}$$ and coplanar with $$\bar{p}$$ and $$\bar{q}$$ is

The value of $$\frac{{ }^{10} \mathrm{C}_{\mathrm{r}}}{{ }^{11} \mathrm{C}_{\mathrm{r}}}$$, when both the numerator and denominator are at their greatest values, is