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Two radioactive materials X_{1} and X_{2} have decay constants 10λ and λ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of X_{1} to that of X_{2} will belie after a time.

#### Options

`1/(10λ)`

`1/(11λ)`

`11/(10λ)`

`1/(9λ)`

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#### Solution

`underline(1/(9λ))`

**Explanation:**

`N_1 = N_0e^{-lambda_1t}`, `N_2 = N_0e^{-lambda_2t}`

∴ `N_1/N_2 = e^{-lambda_1t}/e^{-lambda_2t} = e^{-(lambda_1 - lambda_2)t} = e^{-(10lambda - lambda)t} = e^{-9lambdat}`

Given `N_1/N_2 = e^-1`

∴ -9λt = -1 ⇒ t = `1/(9lambda)`

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