In a radioactive decay chain, ${ }_{90}^{232} Th$ nucleus decays to ${ }_{82}^{212} Pb$ nucleus. Let $N _\alpha$ and $N _\beta$ be the number of $\alpha$ and $\beta^{-}$particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

$(A)$ $N _\alpha=5$  $(B)$ $N _\alpha=6$  $(C)$ $N _\beta=2$  $(D)$ $N _\beta=4$

In a radioactive sample, ${ }_{10}^a K$ nuclei either decay into stable ${ }_{20}^{* 0} Ca$ nuclei with decay constant $4.5 \times 10^{-10}$ per year or into stable ${ }_{18}^{40}$ Ar muclei with decay constant $0.5 \times 10^{-10}$ per year. Given that in this sample all the stable ${ }_{20}^{\infty 0} Ca$ and ${ }_{15}^{20} Ar$ nuclei are produced by the ${ }_{19}^{* 0} K$ muclei only. In time $t \times 10^{\circ}$ years, if the ratio of the sum of stable ${ }_{30}^{40} Ca$ and ${ }_{15} \operatorname{An}$ nuclei to the radioactive ${ }_{19} K$ muclei is $99$ , the ralue of $t$ will be : [Given $\ln 10=2.3]$

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

The activity of a freshly prepared radioactive sample is $10^{10}$ disintegrations per second, whose mean life is $10^9 s$. The mass of an atom of this radioisotope is $10^{-25} kg$. The mass (in $mg$ ) of the radioactive sample is

In Fig. $X$ represents time and $Y$ represent activity of a radioactive sample. Then the activity of sample, varies with time according to the curve

Carbon $ - 14$ decays with half-life of about $5,800\, years$. In a sample of bone, the ratio of carbon $ - 14$ to carbon $ - 12$ is found to be $\frac{1}{4}$ of what it is in free air. This bone may belong to a period about $x$ centuries ago, where $x$ is nearest to