For $0 < x \leq \pi$,$\sinh ^{-1}(\cot x)$ is equal to

In the adjoining figure,three capacitors $C_1, C_2,$ and $C_3$ are connected to a battery. The correct condition will be (Symbols have their usual meanings):

$A$ number $n$ is chosen at random from $\{1, 2, 3, 4, \ldots, 1000\}$. The probability that $n$ is a number that leaves remainder $1$ when divided by $7$ is:

Pollution from animal excreta and organic waste from the kitchen can be most profitably minimised by

$A$ finite ladder is constructed by connecting several sections of $2\,\mu F$ and $4\,\mu F$ capacitor combinations as shown in the figure. It is terminated by a capacitor of capacitance $C$. What value should be chosen for $C$ such that the equivalent capacitance of the ladder between the points $A$ and $B$ becomes independent of the number of sections in between? (Answer in $\mu F$)

Reproducing new plants by cells instead of seeds is known as