To determine the largest distance of an image of a real object from a convex mirror, we need to consider the properties of convex mirrors and the formula for calculating image distance.

Properties of Convex Mirrors:
1. The radius of curvature of a convex mirror is always positive.
2. Convex mirrors always produce virtual, diminished, and upright images.
3. The image formed by a convex mirror is always located behind the mirror.

Formula for Calculating Image Distance:
The formula for calculating the image distance (di) of an object placed in front of a convex mirror is given by:

1/f = 1/do + 1/di

Where:
- f is the focal length of the convex mirror,
- do is the object distance (distance of the object from the mirror),
- di is the image distance (distance of the image from the mirror).

Since the mirror in this question has a radius of curvature of 20 cm, the focal length (f) can be calculated as half the radius of curvature, which is 10 cm.

Largest Distance of the Image:
To find the largest distance of the image, we need to consider the extreme case where the object distance (do) is the greatest. In this case, the object is placed at infinity.

When the object is placed at infinity, the object distance (do) can be considered as very large or approaching infinity.

Using the formula mentioned above, we can substitute do as infinity, which yields:

1/f = 1/infinity + 1/di

As the object distance (do) approaches infinity, the equation simplifies to:

1/f = 0 + 1/di

Since 1/f is a positive value (1/10), the term 1/di must also be positive. For 1/di to be positive, di must be positive.

Therefore, the image distance (di) cannot be zero or negative. It also cannot be infinity because the image formed by a convex mirror is always located behind the mirror.

Hence, the largest distance of an image of a real object from a convex mirror with a radius of curvature of 20 cm is 10 cm, which is option A.