Two radioactive materials X1 and X2 have decay constants 6λ and 3λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2 will be 1 after a time.

The correct option is B 13λ
The ratio of number of nuclei of X1 to that of X2 after time t, N1/N2 = 1
e⁻⁶λt / e⁻³λt = 1
or e⁻³λt = 1
or 3λt = 0
λt = 0 (This is incorrect, there must be a mistake in the problem statement or solution)
Let's assume the question meant to ask when N1/N2 = 1
N1/N2 = N₀e⁻⁶λt / N₀e⁻³λt = 1 (Since initially they have the same number of nuclei, N₀)
e⁻³λt = 1
Taking natural logarithm on both sides:
-3λt = ln(1)
-3λt = 0
t = 0
This implies that the ratio is 1 at time t=0. There's a flaw in either the question or the provided solution. The solution provided seems to have made an error in its derivation. A correct approach would involve solving for t when N1/N2 = 1, given N1 = N₀e⁻⁶λt and N2 = N₀e⁻³λt. However the provided solution has inconsistencies.