(a) Decay constant: Let the number of nuclei of a radioactive substance present at any time be $\mathit{N}$. Decay of number of nuclei $\left(∆\mathit{N}\right)$ in a small interval of time $\left(∆\mathrm{t}\right)$ is proportional to $\mathrm{N}$ and $∆\mathrm{t}$.
$∆\mathrm{N}\propto \mathrm{N}∆\mathrm{t}$

The proportionality constant $\mathit{\lambda }$ is known as the decay constant.
(b) Half-life $=4.5\times {10}^9$ years
Mass number $=238$

$\mathrm{N}=\frac{10\times 6.02\times {10}^{23}}{238}$
Hence, the activity, $\mathrm{A}=\mathit{\lambda }\mathrm{N}$