Two nuclei have their mass numbers in the ratio of $1:3$. The ratio of their densities would be

The equation of the normal to the curve $y^4=ax^3$ at $(a, a)$ is

The current in an $AC$ circuit is given by $i = i_1 \sin \omega t + i_2 \cos \omega t$. Its $r.m.s.$ value is:

When a spring is stretched by $2 \ cm$,it stores $100 \ J$ of energy. If it is further stretched by $2 \ cm$,the stored energy will be increased by $.... \ J$.

$A$ force acts on a $3\, g$ particle in such a way that the position of the particle as a function of time is given by $x = 3t - 4t^2 + t^3$,where $x$ is in $meters$ and $t$ is in $seconds$. The work done during the first $4\, seconds$ is .............. $mJ$.

$A$ cylindrical vessel open at the top is $20 \ cm$ high and $10 \ cm$ in diameter. $A$ circular hole whose cross-sectional area is $1 \ cm^2$ is cut at the centre of the bottom of the vessel. Water flows from a tube above it into the vessel at the rate of $100 \ cm^3 s^{-1}$. The height of water in the vessel under steady state is ....... $cm$ (Take $g = 1000 \ cm s^{-2}$)