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(C) The activity of a radioactive sample is given by $A = |dN/dt| = N\lambda$. Thus,option $A$ is correct.By definition,the decay constant $\lambda$ r

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After a time interval of $1/\lambda$,the number of nuclei remaining in the sample is $N[1 - 1/e]$

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(C) The activity of a radioactive sample is given by $A = |dN/dt| = N\lambda$. Thus,option $A$ is correct.By definition,the decay constant $\lambda$ represents the probability of decay per unit time for a single nucleus. Thus,option $B$ is correct.The number of nuclei remaining after time $t$ is given by $N(t) = N_0 e^{-\lambda t}$.For $t = 1/\lambda$,the number of nuclei remaining is $N(1/\lambda) = N e^{-\lambda(1/\lambda)} = N e^{-1} = N/e$.Option $C$ states that the number of nuclei remaining is $N[1 - 1/e]$,which represents the number of nuclei that have decayed,not the number remaining. Therefore,option $C$ is incorrect.The half-life $T_{1/2}$ is given by $\ln(2)/\lambda$. Thus,option $D$ is correct.

At a given instant,a radioactive sample has $N$ active nuclei. If its decay constant is $\lambda$,which of the following relations is incorrect? (Note: $\lambda$ is very small.)

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