### JEE Main Solved Paper 2017 Question 9

##### Question: A radioactive nucleus A with a half-life T, decays into a nucleus B. At t = 0, there is no nucleus B. At some time t, the ratio of the number of B to that of A is 0.3. Then, t is given by: [JEE Main Solved Paper-2017]

#### Options:

A) $ t=T\log (1.3) $

B) $ t=\frac{T}{\log (1.3)} $

C) $ t=\frac{T\log 2}{2\log 1.3} $

D) $ t=T\frac{\log 1.3}{\log 2} $

## Show Answer

#### Answer:

Correct Answer: D

#### Solution:

- A time t $ \frac{N _{B}}{N _{A}}=.3\Rightarrow N _{B}=.3N _{A} $ also let initially there are total N0 number of nuclei $ N _{A}+N _{B}=N _0 $ $ N _{A}=\frac{N _0}{1.3} $ Also as we know $ N _{A}=N _0{e^{-\lambda t}} $ $ \frac{N _0}{1.3}=N _0{e^{-\lambda t}} $ $ \frac{1}{1.3}={e^{-\lambda t}}\ell n(1.3)=\lambda t $ or $ t=\frac{\ell n(1.3)}{\lambda } $ $ t=\frac{\ell n(1.3)}{\frac{\ell n(2)}{T}}=\frac{\ell n(1.3)}{\ell n(2)}T $