A

To solve this problem, we will use the concept of radioactive decay. The decay rate of a radioactive substance is described by the formula:

$$ N(t) = N_0 e^{-\lambda t} $$

where:
- $N(t)$ is the number of disintegrations at time $t$
- $N_0$ is the initial number of disintegrations
- $\lambda$ is the decay constant

We know the half-life ($T_{1/2}$) of the radioactive element is 1 second. The decay constant $\lambda$ can be calculated using the formula:

$$ \lambda = \frac{\ln(2)}{T_{1/2}} $$

Substituting $T_{1/2} = 1$ second, we get:

$$ \lambda = \ln(2) $$

Given that the initial decay rate $N_0 = 10^3$ disintegrations/sec, let's calculate the decay rate after 1 second and 3 seconds.

**After 1 second:**
$$ N(1) = N_0 e^{-\lambda * 1} = 10^3 e^{-\ln(2)} = \frac{10^3}{2} = 500 $$ disintegrations/sec.

**After 3 seconds:**
$$ N(3) = N_0 e^{-\lambda * 3} = 10^3 e^{-3\ln(2)} = 10^3 \left(\frac{1}{2^3}\right) = 10^3 \cdot \frac{1}{8} = 125 $$ disintegrations/sec.

Therefore, the decay rate after one second is **500 disintegrations/sec** and after three seconds is **125 disintegrations/sec**. The answer is **500 and 125**.