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Question

# A radioactive nucleus A with a half life T, decays into a nucleus B. At t=0, there is no nucleus B. At sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by :

A
t=Tlog(1.3)
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B
t=T2log2log1.3
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C
t=Tlog1.3log2
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D
t=Tlog(1.3)
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Solution

## The correct option is B t=Tlog1.3log2Since the radioactive nucleus has half life ′T′,⇒λ=ln2TA→λBAt t=0, No. of A=N2 and No. of B=0Therefore at time =t,No.of A=N0e−λtNo.of B=N0−N0e−λt=N0(1−e−λt)⇒No.of BNo.of A=N0(1−e−λt)N0eλt=0.3⇒e−λt−1=0.3⇒eλt=1.3⇒λt=ln1.3⇒t=ln1.3λ=Tln1.3ln2=Tlog1.3log2Hence Correct answer is option (C).

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