A radioactive nucleus emits 3 alpha particles and 2 positrons. For the resultant nucleus, the ratio of neutrons to protons is (consider the initial nucleus to have atomic number Z and atomic mass A):

Correct option is B

​$$\text{Given:} \\\text{Initial proton number } Z \\\text{Mass number } A \\\text{Initial neutron number } A - Z \\\text{Emissions:} \\\text{1. Emission of 3 alpha particles:} \\\text{Each alpha particle reduces the atomic number } Z \text{ by 2 and the mass number } A \text{ by 4. So, after 3 alpha emissions:} \\Z = Z - 6 \quad \text{and} \quad A = A - 12 \\\text{2. Emission of 2 positrons:} \\\text{Each positron reduces the atomic number } Z \text{ by 1 and does not affect the mass number } A. \text{ After 2 positron emissions:} \\Z = Z - 8 \quad \text{and} \quad A = A - 12 \\\text{Now, the number of protons is:} \\Z' = Z - 8 \\\text{The number of neutrons is:} \\\text{Neutrons} = A' - Z' = A - 12 - (Z - 8) = A - Z - 4 \\\text{Ratio of neutrons to protons:} \\\frac{\text{Neutrons}}{\text{Protons}} = \frac{A - Z - 4}{Z - 8}$$​​