KVPY SX SB Exam 2012 Question Paper

A radioactive nucleus A has a single decay mode with half life ${\mathit{\tau }}_{\mathit{A}}$. Another radioactive nucleus B has two decay modes 1 and 2. If decay mode 2 were absent, the half life of B would have been $\frac{{\mathit{\tau }}_{\mathit{A}}}{2}$. If decay mode 1 were absent, the half life of B would have been $3{\mathit{\tau }}_{\mathit{A}}$. If the actual half life of B is ${\mathit{\tau }}_{\mathit{B}}$, then the ratio $\frac{{\mathit{\tau }}_{\mathit{B}}}{{\mathit{\tau }}_{\mathit{A}}}$ is