To solve this problem, we need to use the concept of radioactive decay and the decay constant.

Radioactive decay is a process in which the nucleus of an unstable atom loses energy by emitting radiation. The rate at which this decay occurs is measured in disintegrations per unit time, typically per minute or per second.

Let's break down the problem into two parts:

1) Initial disintegration rate:
Given that the disintegration rate at any instant is 5000 disintegrations per minute, we can write this as:
R₀ = 5000 disintegrations/minute

2) Disintegration rate after 5 minutes:
After 5 minutes, the disintegration rate is 1250 disintegrations per minute. We can represent this as:
R₅ = 1250 disintegrations/minute

Now, let's use the equation for radioactive decay to find the decay constant (λ).

The decay constant (λ) is defined as the probability that an atom will decay per unit time. It is given by the equation:
R = R₀ * e^(-λt)

where R is the disintegration rate at time t, R₀ is the initial disintegration rate, λ is the decay constant, and t is the time.

To find the decay constant (λ), we can use the given values for R₀ and R₅ and the time difference of 5 minutes.

Substituting the values into the equation, we get:
R₅ = R₀ * e^(-λ * 5)

Simplifying the equation, we have:
1250 = 5000 * e^(-5λ)

Dividing both sides of the equation by 5000, we get:
0.25 = e^(-5λ)

Taking the natural logarithm (ln) of both sides, we have:
ln(0.25) = ln(e^(-5λ))

Using the property of logarithms, ln(e^(-5λ)) simplifies to:
ln(0.25) = -5λ * ln(e)

Since ln(e) = 1, the equation further simplifies to:
ln(0.25) = -5λ

Now, let's solve for the decay constant (λ).

Dividing both sides of the equation by -5, we get:
λ = ln(0.25) / -5

Simplifying the equation, we have:
λ = - 0.2 * ln(0.25)

Using the property of logarithms, ln(0.25) can also be written as ln(2^(-2)) = -2 * ln(2).

Therefore, the decay constant (λ) is:
λ = -0.2 * (-2) * ln(2) = 0.4 * ln(2)

Hence, the correct answer is option A) 0.4 * ln(2).