The sum of the infinite series ${\tan ^{ - 1}}\left( {\frac{2}{{1 - {1^2} + {1^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{4}{{1 - {2^2} + {2^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{6}{{1 - {3^2} + {3^4}}}} \right) + \dots$ is:

$A$ moving coil galvanometer of resistance $100 \ \Omega$ shows full-scale deflection when a current of $100 \ \mu A$ passes through it. If it is intended to show full-scale deflection when a current of $1 \ mA$ passes through it,the value of shunt resistance in ohms to be connected to the galvanometer is:

In a meter bridge,a $30 \Omega$ resistance is connected in the left gap and a pair of resistances $P$ and $Q$ in the right gap. Measured from the left,the balance point is $37.5 \text{ cm}$ when $P$ and $Q$ are in series,and $71.4 \text{ cm}$ when they are in parallel. The values of $P$ and $Q$ (in $\Omega$) are:

The specific resistance (resistivity) of material $B$ is twice that of material $A$. The diameter of a circular wire made of $B$ is twice that of a wire made of $A$. If the resistances of both wires are equal,then the ratio of their lengths $l_B/l_A$ will be .......

Consider the following reaction equilibrium:
$N_{2(g)} + 3H_{2(g)} \rightleftharpoons 2NH_{3(g)}$
Initially,$1 \text{ mole}$ of $N_2$ and $3 \text{ moles}$ of $H_2$ are taken in a $2 \text{ L}$ flask. At equilibrium state,if the number of moles of $N_2$ is $0.6$,what is the total number of moles of all gases present in the flask?

Which of the following figures represents the variation of particle momentum $p$ and the associated de-Broglie wavelength $\lambda$?