For a substance the average life for $\alpha $ -emission is $1620\ years$ and for $\beta $ emission is $405\ years$ . After how much time the $\frac {1}{4}$ of the material remains by simultaneous emission ………… $years$

Which is the correct expression for half-life

${ }_{92}^{238} U$ is known to undergo radioactive decay to form ${ }_{82}^{206} Pb$ by emitting alpha and beta particles. A rock initially contained $68 \times 10^{-6} g$ of ${ }_{92}^{238} U$. If the number of alpha particles that it would emit during its radioactive decay of ${ }_{92}^{238} U$ to ${ }_{82}^{206} Pb$ in three half-lives is $Z \times 10^{18}$, then what is the value of $Z$?

Deuteron is a bound state of a neutron and a proton with a binding energy $B = 2.2\, MeV$. A $\gamma $ -ray of energy $E$ is aimed at a deuteron nucleus to try to break it into a (neutron + proton) such that the $n$ and $p$ move in the direction of the incident $\gamma $ -ray. If $E = B$, show that this cannot happen. Hence calculate how much bigger than $B$ must $E$ be for such a process to happen. 

An archaeologist analyses the wood in a prehistoric structure and finds that $C^{14}$ (Half life $= 5700\, years$) to $C^{12}$ is only one-fourth of that found in the cells of buried plants. The age of the wood is about ……….$years$