For a radioactive material, its activity A and rate of change of its activity R are defined as A = -dN/dt and R = -dA/dt, where N(t) is the number of nuclei at time t. Two radioactive sources P (mean life τ) and Q (mean life 2τ) have the same activity at t = 0. Their rates of change of activities at t = 2τ are RP and RQ, respectively. If RP/RQ = ne, then the value of n is

The activity of radioactive substance is given as:
A = dN/dt = λN = λN(t=0)e-λt.. (i)
Mean life time τ is related to λ as:
λ = 1/τ.. (ii)
Given activity of P and Q are equal at time t:
λPNP(t=0)e-λPt = λQNQ(t=0)e-λQt.. (iii)
The rate of change of activity can be found by differentiating (i)
dA/dt = λN(t=0)e-λt.. (iv)
Calculating RP and RQ:
RP = λ2PNP(t=0)e-λP(t+2τ)
RQ = λ2QNQ(t=0)e-λP(t+2τ)
RP/RQ = λ2PNP(t=0)e-λPt / λ2QNQ(t=0)e-λQ(t+2τ).. (v)
Equation (v) can be written as:
RP/RQ = λPλPNP(t=0)e-λPte-λP2τ / λQλQNQ(t=0)e-λQte-λQ2τ
From equation (iii):
RP/RQ = λPe-λP2τ / λQe-λQ2τ
RP/RQ = λPe(λQ - λP)2τ / λQ
From equation (i):
λP/λQ = 1/τ / 1/2τ = 2
RP/RQ = 2e(λQ - λP)2τ
RP/RQ = 2e(1/2τ - 1/τ)2τ
RP/RQ = 2e-1 = 2/e ⇒ n = 2