Let $$\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{\mathrm{c}}$$ be a vector such that $$(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=3(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}})$$. If $$\vec{a} \cdot \vec{c}=130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to __________.

If the constant term in the expansion of $$\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$$ is $$\mathrm{p}$$, then $$108 \mathrm{p}$$ is equal to ________.

The number of distinct real roots of the equation $$|x||x+2|-5|x+1|-1=0$$ is __________.

A body of mass $$50 \mathrm{~kg}$$ is lifted to a height of $$20 \mathrm{~m}$$ from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :