At any instant, two elements $X _1$ and $X _2$ have same number of radioactive atoms. If the decay constant of $X _1$ and $X _2$ are $10 \lambda$ and $\lambda$ respectively. then the time when the ratio of their atoms becomes $\frac{1}{e}$ respectively will be

$'Rn$' decays into $'Po'$ by emitting $a -$ particle with half life of $4\, days$. A sample contains $6.4 \times 10^{10}$ atoms of $Rn$. After $12\, days$, the number of atoms of $'Rn'$ left in the sample will be

Half life period of a sample is $15$ years. How long will it take to decay $96.875\%$ of sample ………. $years$

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. A rock initially contained $68 \times 10^{-6} g$ of ${ }_{92}^{238} U$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} U$ to ${ }_{82}^{206} Pb$ in three half-lives is $Z \times 10^{18}$, then what is the value of $Z$?

According to classical physics, $10^{-15}\ m$ is distance of closest approach $(d_c)$ for fusion to occur between two protons. A more accurate and quantum approach says that ${d_c} = \frac{{{\lambda _p}}}{{\sqrt 2 }}$ where $'\lambda _p'$ is de-broglie's wavelength of proton when they were far apart. Using quantum approach, find equation of temperature at centre of star. [Given: $M_p$ is mass of proton, $k$ is boltzman constant]