To determine the minimum size of the fillet weld required to connect two plates of different thicknesses, we need to consider the design requirements and applicable welding codes. Let's break down the solution into the following sections:

1. Understanding Fillet Welds:
A fillet weld is a type of weld that joins two pieces of metal along their edges in a lap joint or a T-joint. It is typically triangular in shape and provides strength and stability to the joint.

2. Weld Size Calculation:
According to welding codes such as the American Welding Society (AWS) D1.1 Structural Welding Code, the minimum fillet weld size is determined based on the thickness of the thinner plate being joined. The required weld size is expressed as a multiple of the plate thickness.

3. Determining the Minimum Weld Size:
In this case, we have two plates with thicknesses of 10 mm and 12 mm. Since the minimum weld size is determined by the thinner plate, we use 10 mm as the reference.

According to the AWS D1.1 code, for plates up to 13 mm thick, the minimum fillet weld size is given by the formula:
Minimum Weld Size = 0.7 x Plate Thickness

Substituting the plate thickness of 10 mm into the formula:
Minimum Weld Size = 0.7 x 10 mm = 7 mm

However, it is important to note that the code specifies a minimum weld size of 3 mm for plates less than 4.8 mm thick. Therefore, even though the calculated minimum weld size is 7 mm, the actual minimum weld size required is the larger value between 7 mm and 3 mm.

4. Final Answer:
Comparing the calculated minimum weld size of 7 mm with the specified minimum weld size of 3 mm, we can conclude that the minimum size of the fillet weld required to connect the two plates of thickness 10 mm and 12 mm is 7 mm.

Therefore, the correct answer is option C) 5 mm.