In a $P$-type semiconductor,germanium is doped with:

If $A=35^{\circ}, B=15^{\circ}$ and $C=40^{\circ}$,then $\tan A \cdot \tan B+\tan B \cdot \tan C+\tan C \cdot \tan A$ is equal to

If the axes are rotated through an angle $45^{\circ}$ in the positive direction without changing the origin,then the coordinates of the point $(\sqrt{2}, 4)$ in the old system are

$A$ vertical pole subtends an angle $\tan ^{-1}\left(\frac{1}{2}\right)$ at a point $P$ on the ground. If the angles subtended by the upper half and the lower half of the pole at $P$ are respectively $\alpha$ and $\beta$,then $(\tan \alpha, \tan \beta)$ is equal to

The following reaction is an example of a ....... reaction. $CH_2Br-CH_2Br + 2KOH(alc.) \xrightarrow{\Delta} CH \equiv CH + 2KBr + 2H_2O$

If the direction ratios $(l, m, n)$ of two lines satisfy the equations $l+m+n=0$ and $mn-2ln+lm=0$,then the angle between the lines is