State the relation between average life and decay constant. 

A parent nucleus $X$ is decaying into daughter nucleus $Y$ which in turn decays to $Z$. The half lives of $X$ and $Y$ are $40000 \,yr$ and $20 \,yr$, respectively. In a certain sample, it is found that the number of $Y$ nuclei hardly changes with time. If the number of $X$ nuclei in the sample is $4 \times 10^{20}$, the number of $Y$ nuclei present in it is

A radioactive sample has ${N_0}$ active atoms at $t = 0$. If the rate of disintegration at any time is $R$ and the number of atoms is $N$, then the ratio $ R/N$ varies with time as

A radioactive sample consists of two distinct species having equal number of atoms $N_0$ initially. The mean-life of one species is $\tau $ and of the other is $5\tau $. The decay products in both cases is stable. The total number of radioactive nuclei at $t = 5\tau $ is

The half life of a radioactive element which has only $\frac{1}{{32}}$ of its original mass left after a lapse of $60\, days$ is ........$days$

After absorbing a slowly moving neutron of mass $m_N$  $(momentum $~ $0)$ a nucleus of mass $M$ breaks into two nuclei of masses $m_1$ and $3m_1$ $(4m_1 = M + m_N)$, respectively. If the de Broglie wavelength of the nucleus with mass $m_1$ is $\lambda$, then de Broglie wavelength of the other nucleus will be