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13.Nuclei
hard
Activities of three radioactive substances $A , B$ and $C$ are represented by the curves $A, B$ and $C,$ in the figure. Then their half-lives $T _{\frac{1}{2}}( A ): T _{\frac{1}{2}}( B ): T _{\frac{1}{2}}( C )$ are in the ratio

A
$3: 2: 1$
B
$4: 3: 1$
C
$2: 1: 3$
D
$2: 1: 1$
(JEE MAIN-2020)
Solution
$R = R _{0} e ^{-\lambda t}$
ln $R =\ln R _{0}-\lambda t$
$\lambda_{ A }=\frac{6}{10} \Rightarrow T _{ A }=\frac{10}{6}$ ln $2$
$\lambda_{ B }=\frac{6}{5} \Rightarrow T _{ B }=\frac{5 \ln 2}{6}$
$\lambda_{ C }=\frac{2}{5} \Rightarrow T _{ C }=\frac{5 \ln 2}{2}$
$\frac{10}{6}: \frac{5}{6}: \frac{15}{6}:: 2: 1: 3$
Standard 12
Physics