**Core Concept**
The question is testing the ability to diagnose a specific condition characterized by recurrent pyogenic infections due to encapsulated bacteria. This condition is associated with a genetic defect in the complement system, specifically the C3 convertase enzyme, leading to impaired opsonization and phagocytosis of encapsulated pathogens.

**Why the Correct Answer is Right**
The correct answer involves investigating the function of the complement system, particularly the alternative pathway. In this condition, the boy's immune system is unable to effectively clear encapsulated bacteria due to a deficiency in the C3 convertase enzyme, leading to recurrent infections. This deficiency can be assessed through a serum complement level study, specifically measuring the alternative pathway C3 convertase activity.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is not relevant to the diagnosis of the condition described. It may be a test for a different condition or a general screening test, but it does not directly address the specific problem of recurrent pyogenic infections due to encapsulated bacteria.
* **Option B:** This option may be a test for a different type of immune deficiency or a general assessment of the boy's immune system, but it does not specifically address the issue of impaired complement function.
* **Option C:** This option may be related to the diagnosis of a different condition or a general assessment of the boy's overall health, but it does not directly address the specific problem of recurrent pyogenic infections due to encapsulated bacteria.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that recurrent pyogenic infections due to encapsulated bacteria can be a clue to a genetic defect in the complement system, particularly the alternative pathway. This is a classic exam trap, and being aware of this association can help you arrive at the correct diagnosis.

**Correct Answer:** C. Serum complement level study.