**Introduction:**

In physics, the center of mass is a point that represents the average position of all the particles in a system. It is often used to simplify calculations and analyze the motion of an object. In this question, we are given that the center of mass of a body is located at the origin, and we need to determine the possible x-coordinates of the particles in the body.

**Explanation:**

a) May be all positive:
- If all the particles in the body have positive x-coordinates, then the center of mass would not be at the origin. The x-coordinate of the center of mass is determined by taking the average of the x-coordinates of all the particles. If all the x-coordinates are positive, then the average will also be positive, and the center of mass will have a positive x-coordinate.

b) May be all negative:
- Similarly, if all the particles have negative x-coordinates, the center of mass would not be at the origin. The x-coordinate of the center of mass is determined by taking the average of the x-coordinates of all the particles. If all the x-coordinates are negative, then the average will also be negative, and the center of mass will have a negative x-coordinate.

c) May be all non-negative:
- If all the particles have non-negative x-coordinates, then it is possible for the center of mass to be at the origin. This is because the average of non-negative numbers can be zero. For example, if there are two particles with x-coordinates +2 and -2, the average is (2 + (-2))/2 = 0, and the center of mass will be at the origin.

d) May be positive for some cases and negative in other cases:
- It is also possible for some particles to have positive x-coordinates and others to have negative x-coordinates. In this case, the x-coordinate of the center of mass will depend on the distribution and mass of the particles. If the positive x-coordinate particles are more massive or closer to the origin than the negative x-coordinate particles, the center of mass will have a positive x-coordinate. Conversely, if the negative x-coordinate particles are more massive or closer to the origin than the positive x-coordinate particles, the center of mass will have a negative x-coordinate.

**Conclusion:**

In conclusion, the x-coordinates of the particles in a body with its center of mass at the origin can be all positive, all negative, all non-negative, or a combination of positive and negative depending on the distribution and mass of the particles. Therefore, the correct answers are options C and D.