**JEE Advanced 2019 Paper 1, Question 4**

In a radioactive sample, nuclei decay into stable nuclei with decay constant per year or into stable nuclei with decay constant per year. In this sample all the stable and nuclei are produced by the nuclei only. In time years, if the ratio of the sum of stable and nuclei to the radioactive nuclei is 99, the value of will be [Given ],

- 1.15
- 9.2
- 2.3
- 4.6

**Solution**

Let us first consider the situation where a sample of nuclei decays into a *single* type of nuclei . The number of nuclei changes with time according to the equation

(1)

where is the decay constant. It is the inverse of the average lifetime of the nuclei before it decays into . So, is the probability that a single nuclei decays in the small time interval . When we multiply this with the total number of nuclei available we get , the change in the number of nuclei in that that time. The negative sign in the RHS accounts for the fact that decreases over time.

Coming to the question at hand, we’re told that the nuclei can decay into or . Let us denote the number of nuclei as . Reasoning as we did above we can see that the probability for the two decays are and . So the total probability that a nuclei decays in time is obtained by adding these two probabilities. Therefore, the change in over a small time interval must be

(2)

That is,

(3)

which has the solution

(4)

where per year and is the initial number of nuclei at time . In the last line of the problem we’re told that

at time years. Substituting this into (4) and simplifying,

(5)