What is the half-life (in years) period of a radioactive material if its activity drops to $1 / 16^{\text {th }}$ of its initial value of $30$ years?

A certain radioactive nuclide of mass number $m_x$ disintegrates, with the emission of an electron and $\gamma$ radiation only, to give second nuclied of mass number $m_y.$ Which one of the following equation correctly relates $m_x$ and $m_y$ ?

For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A=-\frac{d N}{d t}$ and $R=-\frac{d A}{d t}$, where $N(t)$ is the number of nuclei at time $t$. Two radioactive sources $P$ (mean life $\tau$ ) and $Q$ (mean life $2 \tau$ ) have the same activity at $t=0$. Their rates of change of activities at $t=2 \tau$ are $R_p$ and $R_Q$, respectively. If $\frac{R_p}{R_Q}=\frac{n}{e}$, then the value of $n$ is

A sample of radioactive element containing $4 \times 10^{16}$ active nuclei. Half life of element is $10$ days, then number of decayed nuclei after $30$ days is  …….. $\times 10^{16}$ 

After $3$ hours, only $0.25 \,mg$ of a pure radioactive material is left. If initial mass was $2 \,mg$ then the half life of the substance is …… $hr$