If $\vec{P} \cdot \vec{Q} = PQ$,then the angle between $\vec{P}$ and $\vec{Q}$ is ............. $^\circ$.

Particles and their anti-particles have

An observer is standing $500 ~m$ away from a vertical hill. Starting between the observer and the hill,a police van sounding a siren of frequency $1000 ~Hz$ moves towards the hill with a uniform speed. If the frequency of the sound heard directly from the siren is $970 ~Hz$,the frequency of the sound heard after reflection from the hill (in $Hz$) is about,(velocity of sound $= 330 ~m/s$):

The quantum number which explains the line spectra observed as doublets in the case of hydrogen and alkali metals and doublets and triplets in the case of alkaline earth metals is

Consider the following reaction: $C_6H_5NO_2$ $\xrightarrow{Sn/HCl} X$ $\xrightarrow{C_6H_5COCl} Y + HCl$. What is $Y$?

$\frac{1}{\sqrt{11 - 2\sqrt{30}}} - \frac{3}{\sqrt{7 - 2\sqrt{10}}} - \frac{4}{\sqrt{8 + 4\sqrt{3}}} = $