When $_{90}Th^{238}$ transforms into $_{83}Bi^{222}$,the number of $\alpha$ and $\beta$ particles emitted are:

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 $\beta^{-}$ decay, a neutron transforms into a proton within the nucleus according to the equation:
$\text{neutron} \rightarrow \text{proton} + \beta^{-} + x$
In this equation, the particle represented by '$x$' is:

Some radioactive nucleus may emit

For the $\beta^{+}$ (positron) emission from a nucleus,there is another competing process known as electron capture (an electron from an inner orbit,say,the $K$-shell,is captured by the nucleus and a neutrino is emitted).
$_{z}^{A} X + e^{-} \rightarrow _{z-1}^{A} Y + \nu$
Show that if $\beta^{+}$ emission is energetically allowed,electron capture is necessarily allowed,but not vice-versa.

The intensity of gamma rays is reduced to half when they pass through a lead sheet of thickness $2.5 \, mm$. The absorption coefficient of lead is ......... $mm^{-1}$.