There are two radioactive substances A and B . Decay constant of B...

Let ${\lambda}_{A} = \lambda ∴ {\lambda}_{B} = 2\lambda$

If ${N}_{0}$ is total number of atoms in $A$ and $B$ at $t = 0$, then initial rate of disintegration of $A = \lambda{N}_{0}$, and initial rate of disintegration of $B = 2\lambda{N}_{0}$

As ${\lambda}_{B} = 2{\lambda}_{A}$

$∴ {T}_{B} = \dfrac{1}{2}{T}_{A}$

$ie,$half-life of $B$ is half the half-life of $A$.

After one half-life of $A$

${\left(- \dfrac{dN}{dt}\right)}_{A} = \dfrac{\lambda{N}_{0}}{2}$

Equivalently, after two half lives of $B$

${\left(- \dfrac{dN}{dt}\right)}_{B} = \dfrac{2\lambda{N}_{0}}{4} = \dfrac{\lambda{N}_{0}}{2}$

Clearly, ${\left(- \dfrac{dN}{dt}\right)}_{A} = - {\left(\dfrac{dN}{dt}\right)}_{B}$

after $n = 1ie,$one half-life of $A$