There are three lumps of a given radioactive substance. Their activity is in the ratio of $1 : 2 : 3$ now. What will be the ratio of their activities at any further date?

There are two radionuclei $A$ and $B.$ $A$ is an alpha emitter and $B$ is a beta emitter. Their disintegration constants are in the ratio of $1 : 2.$ What should be the ratio of the number of atoms of the two at time $t = 0$ so that the probabilities of getting $\alpha$ and $\beta$ particles are the same at time $t = 0$?

$A$ radioisotope $X$ with a half-life of $1.4 \times 10^{9} \text{ years}$ decays into $Y$,which is stable. $A$ sample of rock from a cave was found to contain $X$ and $Y$ in the ratio $1:7$. The age of the rock is ........ $\times 10^{9} \text{ years}$.

The average life $T$ and the decay constant $\lambda$ of a radioactive nucleus are related as

Given below are two statements:
Statement $I$: The law of radioactive decay states that the number of nuclei undergoing the decay per unit time is directly proportional to the total number of nuclei in the sample.
Statement $II$: The half-life of a radionuclide is the time required for the number of radioactive nuclei to reduce to half of its initial value at time $t = 0$.
In the light of the above statements, choose the most appropriate answer from the options given below:

The half-lives of $X$ and $Y$ are $3 \ min$ and $27 \ min$ respectively. If they have the same activity at a certain instant,find the ratio of the number of excited nuclei of $X$ and $Y$ at that instant.