The initial activity of a certain radioactive isotope was measured as $16000\ counts\  min^{-1}$. Given that the only activity measured was due to this isotope and that its activity after $12\, h$ was $2000\ counts\ min^{-1}$, its half-life, in hours, is nearest to

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)

The decay constant of the end product of a radioactive series is

The fraction $f$ of radioactive material that has decayed in time $t$, varies with time $t$. The correct variation is given by the curve

The half-life of a radioactive element $A$ is the same as the mean-life of another radioactive element $B.$ Initially both substances have the same number of atoms, then

Assertion : If the half life of a radioactive substance is $40\, days$ then $25\%$ substance decay in $20\, days$

Reason : $N = {N_0}\,{\left( {\frac{1}{2}} \right)^n}$

where, $n = \frac{{{\rm{time\, elapsed}}}}{{{\rm{half \,life \,period}}}}$