A radioactive nucleus emits 4 $\alpha$ -particles and 7 $\beta$-particles in succession. The ratio of number of neutrons  of that of protons is [A= mass number, Z= atomic number]

A) $\frac{A-Z-13}{Z-2}$

B) $\frac{A-Z-15}{Z-1}$

C) $\frac{A-Z-13}{Z-1}$

D) $\frac{A-Z-11}{Z-2}$


Option B


 Let us  assume, a particle X having atomic number Z and mass  number A.

 when an $\alpha $- particle is emitted by a nucleus, then its atomic number decreases by 2 and mass number decreases by 4 . So  , for given case,

$_{Z}X^{A}$   $\underrightarrow{4\alpha-particle}$    $_{Z-8}Y^{A-16}$

 when a  $\beta$- particle is emitted by a nucleus its atomic  number increases by one and mass number remains  unchanged. So, for given case,

$_{Z-8}Y^{A-16}$  $\underrightarrow{7-\beta particle}$  $_{Z-1}Y^{A-16}$

 $\therefore$    Number of neutrons/ number of protons =   $\frac{(A-16)-(Z-1)}{(Z-1)}$

  =  $\frac{A-Z-15}{(Z-1)}$