$\sin 120^{\circ} \cos 150^{\circ} - \cos 240^{\circ} \sin 330^{\circ}$ is equal to :

An elementary particle of mass $m$ and charge $+e$ is projected with velocity $v$ at a much more massive particle of charge $Ze$,where $Z > 0$. What is the closest possible approach of the incident particle?

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$,then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

At a certain temperature and at infinite dilution,the equivalent conductances of sodium benzoate,hydrochloric acid,and sodium chloride are $240$,$349$,and $229 \ \Omega^{-1} \ cm^2 \ \text{equiv}^{-1}$ respectively. The equivalent conductance of benzoic acid in $\Omega^{-1} \ cm^2 \ \text{equiv}^{-1}$ at the same conditions is:

$A$ square loop of area $25 \, cm^2$ has a resistance of $10 \, \Omega$. This loop is placed in a uniform magnetic field of magnitude $40 \, T$. The plane of the loop is perpendicular to the magnetic field. The work done in pulling the loop out of the magnetic field slowly and uniformly in one second will be:

$A$ charge $q$ is placed at the centre of the line joining two equal charges $Q$. The system of the three charges will be in equilibrium if $q$ is equal to: