If the line $$x-2 y=\mathrm{m}(\mathrm{m} \in \mathrm{Z})$$ intersects the circle $$x^2+y^2=2 x+4 y$$ at two distinct points, then the number of possible values of $m$ are

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors such that $$|\bar{a}+\bar{b}+\bar{c}|=1, \overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})$$ and $$|\overline{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overline{\mathrm{b}}|=\frac{1}{\sqrt{2}},|\overline{\mathrm{c}}|=\frac{1}{\sqrt{6}}$$, then the angle between $$\bar{a}$$ and $$\bar{b}$$ is

The equation $$x^3+x-1=0$$ has

Let $$\bar{a}, \bar{b}, \bar{c}$$ be three vectors such that $$|\bar{a}|=\sqrt{3}, |\bar{b}|=5, \bar{b} \cdot \bar{c}=10$$ and the angle between $$\bar{b}$$ and $$\bar{c}$$ is $$\frac{\pi}{3}$$. If $$\bar{a}$$ is perpendicular to the vector $$\bar{b} \times \bar{c}$$, then $$|\bar{a} \times(\bar{b} \times \bar{c})|$$ is equal to