At time $t=0$ some radioactive gas is injected into a sealed vessel. At time $T$ some more of the gas is injected into the vessel. Which one of the following graphs best represents the logarithm of the activity $A$ of the gas with time $t$ ?

A sample originally contaived $10^{20}$ radioactive atoms, which emit $\alpha -$ particles. The ratio of $\alpha -$ particles emitted in the third year to that emitted during the second year is $0.3.$ How many $\alpha -$ particles were emitted in the first year?

Ther percentage of ${ }^{235} U$ presently on earth is $0.72$ and the rest $(99.28 \%)$ may be taken to be ${ }^{233} U$. Assume that all uranium on earth was produced in a supernova explosion long ago with the initial ratio ${ }^{235} U /^{335} U =2.0$. How long ago did the supernova event occur? (Take the half-lives of ${ }^{235} U$ and ${ }^{238} U$ to be $7.1 \times 10^5$ years and $4.5 \times 10^{9}$ years respectively)

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

In a radioactive substance at $t = 0$, the number of atoms is $8 \times {10^4}$. Its half life period is $3$ years. The number of atoms $1 \times {10^4}$ will remain after interval ...........$years$

The half-life of radioactive Polonium $(Po)$ is $138.6$ days. For ten lakh Polonium atoms, the number of disintegrations in $24$ hours is