**Core Concept**
The patient's symptoms of headache and flushing, combined with a family history of thyroid tumor, suggest a possible diagnosis of Multiple Endocrine Neoplasia Type 2B (MEN2B), a rare genetic disorder characterized by the development of medullary thyroid carcinoma, pheochromocytoma, and other endocrine tumors.

**Why the Correct Answer is Right**
The investigation of choice for this patient would be a genetic test to detect the RET proto-oncogene mutation, which is responsible for MEN2B. This mutation leads to the activation of the RET tyrosine kinase receptor, resulting in the uncontrolled growth of cells in the thyroid gland and other endocrine tissues. A positive family history of thyroid tumor increases the likelihood of this genetic mutation.

**Why Each Wrong Option is Incorrect**
* **Option A:** A thyroid function test (TFT) would not be the primary investigation for this patient, as the symptoms and family history suggest a structural or genetic issue rather than a functional thyroid disorder.
* **Option B:** A chest X-ray or CT scan may be useful in evaluating the patient's symptoms, but it would not be the initial investigation of choice for a suspected genetic disorder like MEN2B.
* **Option C:** A fasting glucose test or insulin tolerance test may be relevant for patients with endocrine disorders, but it is not directly related to the suspected diagnosis of MEN2B.

**Clinical Pearl / High-Yield Fact**
The RET proto-oncogene mutation is a key genetic marker for MEN2B, and early detection through genetic testing can lead to the prompt treatment of medullary thyroid carcinoma and other associated endocrine tumors.

**Correct Answer:** A. Genetic test for RET proto-oncogene mutation.