In the Liebermann's nitroso reaction,the sequential changes in the colour of phenol occur as:

$A$ transistor-oscillator using a resonant circuit with an inductor $L$ (of negligible resistance) and a capacitor $C$ in series produces oscillations of frequency $f$. If $L$ is doubled and $C$ is changed to $4C$,the frequency will be

The probability that a number selected at random from the set of numbers ${1, 2, 3, \ldots, 100}$ is a cube, is

In the following compounds,the order of acidity is:
$(I)$ Phenol
$(II)$ $p$-Cresol ($4$-methylphenol)
$(III)$ $m$-Nitrophenol
$(IV)$ $p$-Nitrophenol

$\int (1+x-x^{-1}) e^{x+x^{-1}} dx$ is equal to :

An infinitely long conductor $PQR$ is bent to form a right angle as shown. $A$ current $I$ flows through $PQR$. The magnetic field due to this current at the point $M$ is $H_1$. Now,another infinitely long straight conductor $QS$ is connected at $Q$ such that the current in $PQ$ remains unchanged. The magnetic field at $M$ is now $H_2$. The ratio $H_1/H_2$ is given by