**Core Concept**
The patient's presentation of weight gain, oligomenorrhea, amenorrhea, and hypertension suggests a possible case of polycystic ovary syndrome (PCOS) with potential secondary hypertension due to insulin resistance.

**Why the Correct Answer is Right**
The most appropriate investigation for this patient is a glucose tolerance test (GTT) to assess for impaired glucose regulation and insulin resistance, which are key features of PCOS. This test will help identify the underlying metabolic abnormalities contributing to her symptoms and hypertension. Insulin resistance is a common finding in PCOS patients, and it can lead to secondary hypertension through various mechanisms, including increased sympathetic tone and vascular dysfunction.

**Why Each Wrong Option is Incorrect**

* **Option A:** FSH/LH ratio is a useful diagnostic tool in PCOS, but it does not directly address the patient's hypertension or metabolic abnormalities. While it can help confirm a diagnosis of PCOS, it is not the most appropriate initial investigation in this case.
* **Option B:** Ultrasound is useful for identifying ovarian morphology in PCOS, but it does not directly assess the patient's metabolic or cardiovascular risk factors. While ovarian cysts are a common feature of PCOS, they do not directly contribute to hypertension.
* **Option C:** Androgen levels can be elevated in PCOS, but they do not directly address the patient's hypertension or metabolic abnormalities. While androgen excess can contribute to symptoms such as hirsutism, it is not the most relevant factor in this case.
* **Option D:** Thyroid function tests (TFTs) are important to rule out thyroid dysfunction as a cause of secondary amenorrhea, but they are not the most appropriate initial investigation in this case. The patient's presentation is more suggestive of PCOS than thyroid dysfunction.

**Clinical Pearl / High-Yield Fact**
PCOS is a common endocrine disorder in women of reproductive age, affecting approximately 5-10% of women. It is often associated with insulin resistance, obesity, and hypertension, making early diagnosis and management crucial to prevent long-term cardiovascular and metabolic complications.

**Correct Answer:** C. Glucose tolerance test (GTT)