Let $$a_{1}, a_{2}, \ldots, a_{n}$$ be in A.P. If $$a_{5}=2 a_{7}$$ and $$a_{11}=18$$, then

$$12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$$ is equal to ____________.

Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$$ and $$|\vec{a} \times \vec{b}|=\sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^{2}$$ is equal to ___________.

Let for $$x \in \mathbb{R}$$,

$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. \text {. } $$

Then area bounded by the curve $$y=(f \circ g)(x)$$ and the lines $$y=0,2 y-x=15$$ is equal to __________.

Let $$\theta$$ be the angle between the planes $$P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$$ and $$P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$$. Let $$\mathrm{L}$$ be the line that meets $$P_{2}$$ at the point $$(4,-2,5)$$ and makes an angle $$\theta$$ with the normal of $$P_{2}$$. If $$\alpha$$ is the angle between $$\mathrm{L}$$ and $$P_{2}$$, then $$\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$$ is equal to ____________.