Starting with a sample of pure $^{66}Cu,\,\frac{7}{8}$ of it decays into $Zn$ in $15\, min$. The corresponding half-life is .......... $min$

A radioactive material decays by simultaneous emission of two particles with respective half lives $1620$ and $810$ years. The time (in years) after which one- fourth of the material remains is

The graph represents the decay of a newly prepared sample of radioactive nuclide $X$ to a stable nuclide $Y$ . The half-life of $X$ is $\tau $ .  The growth curve for $Y$ intersects the decay curve for $X$ after time $T$ . What is the time $T$ ?

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

If half life of a radioactive element is $3\, hours$, after $9\, hours$ its activity becomes

The half-life of a radioactive element $A$ is the same as the mean-life of another radioactive element $B.$ Initially both substances have the same number of atoms, then