If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are three coplanar vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}=8$ and the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $45^{\circ}$ then the value of $|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|$ is

If $u=\log (\sqrt{x-1}-\sqrt{x+1})$ and $v=\sqrt{x+1}+\sqrt{x-1}$ then $\frac{d u}{d v}=\ldots$.

The line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z}{1}$ intersects the XY plane and the YZ plane at points A and B respectively. The equation of line through the points A and B is

A fair $n$ faced die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $\mathrm{n}=($ where $\mathrm{n} \in \mathbb{N})$