A common tangent $$\mathrm{T}$$ to the curves $$\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ and $$C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$$ does not pass through the fourth quadrant. If $$\mathrm{T}$$ touches $$\mathrm{C}_{1}$$ at $$\left(x_{1}, y_{1}\right)$$ and $$\mathrm{C}_{2}$$ at $$\left(x_{2}, y_{2}\right)$$, then $$\left|2 x_{1}+x_{2}\right|$$ is equal to ______________.

Let $$\overrightarrow a $$, $$\overrightarrow b $$, $$\overrightarrow c $$ be three non-coplanar vectors such that $$\overrightarrow a $$ $$\times$$ $$\overrightarrow b $$ = 4$$\overrightarrow c $$, $$\overrightarrow b $$ $$\times$$ $$\overrightarrow c $$ = 9$$\overrightarrow a $$ and $$\overrightarrow c $$ $$\times$$ $$\overrightarrow a $$ = $$\alpha$$$$\overrightarrow b $$, $$\alpha$$ > 0. If $$\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| + \left| {\overrightarrow c } \right| = {1 \over {36}}$$, then $$\alpha$$ is equal to __________.

An expression of energy density is given by $$u=\frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$$, where $$\alpha, \beta$$ are constants, $$x$$ is displacement, $$k$$ is Boltzmann constant and t is the temperature. The dimensions of $$\beta$$ will be :

A body of mass $$10 \mathrm{~kg}$$ is projected at an angle of $$45^{\circ}$$ with the horizontal. The trajectory of the body is observed to pass through a point $$(20,10)$$. If $$\mathrm{T}$$ is the time of flight, then its momentum vector, at time $$\mathrm{t}=\frac{\mathrm{T}}{\sqrt{2}}$$, is _____________.

[Take $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$$ ]