**Core Concept**
The question is testing the student's ability to recognize the clinical presentation of tuberculosis (TB) affecting the larynx, which can cause hoarseness of voice and painful swallowing due to inflammation and scarring of the vocal cords.

**Why the Correct Answer is Right**
TB larynx, also known as laryngeal tuberculosis, is a rare but serious form of extrapulmonary tuberculosis that affects the larynx. The patient's symptoms of hoarseness of voice and painful swallowing, combined with the presence of sputum AFB (Acid-Fast Bacilli) positivity, suggest that the TB is extrapulmonary. The examination finding of mamillated adenoids and hyperaemia on the vocal cord is consistent with laryngeal TB, which can cause ulceration, scarring, and inflammation of the vocal cords. This leads to hoarseness of voice and painful swallowing.

**Why Each Wrong Option is Incorrect**
**Option A:** Rinke's edema is a condition that affects the vocal cords, causing edema and hoarseness of voice, but it is not associated with sputum AFB positivity or the presence of mamillated adenoids and hyperaemia.

**Option B:** Epiglottitis is an inflammation of the epiglottis, which can cause painful swallowing and difficulty breathing, but it is not typically associated with sputum AFB positivity or the specific examination findings of mamillated adenoids and hyperaemia on the vocal cord.

**Option D:** Cancer of the larynx can cause hoarseness of voice and painful swallowing, but it is not typically associated with sputum AFB positivity or the presence of mamillated adenoids and hyperaemia on the vocal cord.

**Clinical Pearl / High-Yield Fact**
Laryngeal TB can be a challenging diagnosis to make, as the symptoms can be non-specific and the examination findings may be subtle. However, the presence of sputum AFB positivity and the specific examination findings of mamillated adenoids and hyperaemia on the vocal cord can suggest laryngeal TB as the diagnosis.

**✓ Correct Answer: C. TB larynx**