$A$ satellite of mass $m$,initially at rest on the Earth,is launched into a circular orbit at a height equal to the radius of the Earth. The minimum energy required is

$A$ battery is used to charge a parallel plate capacitor until the potential difference between the plates becomes equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor to the work done by the battery is

The nucleus of an element contains $9$ protons. Its valency would be

If $200 \text{ MeV}$ of energy is released in the fission of one nucleus of ${ }_{92}^{235} \text{U}$,then the number of nuclei that must undergo fission to release an energy of $1000 \text{ J}$ is

$A$ and $B$ are two radioactive substances whose half-lives are $1$ and $2$ years respectively. Initially, $10 \, g$ of $A$ and $1 \, g$ of $B$ are taken. The time (approximate) after which they will have the same quantity remaining is ........... $years$.

If $\frac{\cos x}{\cos (x-2y)} = \lambda$,then $\tan (x-y) \tan y$ is equal to