**Explanation:**

The new Cartesian sign convention is a set of rules used in geometric optics to determine the sign of various quantities involved in the reflection and refraction of light. These conventions are based on the principles of ray optics and help in simplifying calculations and determining the nature of images formed by mirrors and lenses.

**Focal Length of Convex Mirror:**

In the new Cartesian sign convention, the focal length of a convex mirror is considered positive. This is because a convex mirror is a diverging mirror, which means that it causes parallel rays of light to diverge after reflection. The focal point of a convex mirror is virtual and located behind the mirror. By convention, the focal length is taken as positive for a convex mirror.

**Focal Length of Concave Mirror:**

In contrast, the focal length of a concave mirror is considered negative in the new Cartesian sign convention. A concave mirror is a converging mirror that brings parallel rays of light to a focus after reflection. The focal point of a concave mirror is real and located in front of the mirror. By convention, the focal length is taken as negative for a concave mirror.

**Sign Convention for Distances:**

In addition to the focal length, the new Cartesian sign convention also defines the sign convention for distances. According to this convention:

- Distances measured in the direction of incident light are considered positive.
- Distances measured in the direction opposite to the incident light are considered negative.

**Application of the Sign Convention:**

Using the new Cartesian sign convention, we can determine the nature of the image formed by a mirror based on the positions of the object and the image.

For a convex mirror:
- When the object is real and located in front of the mirror, the image formed is virtual, erect, and diminished.
- When the object is at infinity, the image formed is a point at the focal point of the mirror.
- When the object is virtual and located behind the mirror, the image formed is virtual, erect, and enlarged.

For a concave mirror:
- When the object is real and located beyond the center of curvature, the image formed is real, inverted, and diminished.
- When the object is at the center of curvature, the image formed is real, inverted, and of the same size.
- When the object is real and located between the center of curvature and the focal point, the image formed is real, inverted, and enlarged.
- When the object is at the focal point, the reflected rays become parallel and no image is formed.
- When the object is virtual and located behind the mirror, the image formed is virtual, erect, and enlarged.

Therefore, based on the new Cartesian sign convention, the correct answer is option 'D'. The focal length of a convex mirror is positive and that of a concave mirror is negative.