**Core Concept**
The patient presents with signs of Cushing's syndrome, including weight gain, oligomenorrhoea, and hypertension. Cushing's syndrome is a disorder caused by excess levels of cortisol in the body, which can be due to various causes including endogenous production (e.g., adrenal tumors) or exogenous administration (e.g., glucocorticoid therapy).

**Why the Correct Answer is Right**
The most appropriate initial investigation for suspected Cushing's syndrome is the measurement of plasma cortisol. Elevated cortisol levels can be due to various causes, but the next step in diagnosis would be to differentiate between endogenous and exogenous sources. Plasma cortisol can be measured at different times of the day, and in combination with other tests, can help diagnose Cushing's syndrome. In this case, the patient's presentation with weight gain, oligomenorrhoea, and hypertension suggests Cushing's syndrome.

**Why Each Wrong Option is Incorrect**

**Option A:** Serum electrolytes are not the most appropriate initial investigation for suspected Cushing's syndrome. While electrolyte imbalances can be seen in Cushing's syndrome due to hypokalemic alkalosis, measuring serum electrolytes does not directly diagnose the condition.

**Option C:** Plasma testosterone and ultrasound are not the most appropriate initial investigations for suspected Cushing's syndrome. While an ultrasound can help identify adrenal tumors, measuring plasma testosterone is more relevant in the evaluation of polycystic ovary syndrome (PCOS) or hyperandrogenism.

**Option D:** T3, T4, and TSH are not the most appropriate initial investigations for suspected Cushing's syndrome. These tests are used in the evaluation of thyroid disorders, and while thyroid dysfunction can be seen in Cushing's syndrome, it is not the primary concern in this case.

**Clinical Pearl / High-Yield Fact**
In cases of suspected Cushing's syndrome, it's essential to remember that the diagnosis can be challenging, and a high index of suspicion is required. A combination of clinical evaluation, laboratory tests, and imaging studies is often necessary to establish the diagnosis.

**✓ Correct Answer: B. Plasma cortisol**