Introduction:
Cockroaches are terrestrial insects that belong to the class Insecta and phylum Arthropoda. Like all living organisms, cockroaches produce metabolic waste that needs to be eliminated from their bodies. The process of eliminating waste is known as excretion. In cockroaches, excretion primarily occurs through the haemolymph, which is the insect equivalent of blood. The haemolymph transports waste products throughout the body for eventual elimination.

Explanation:
The nitrogenous waste produced by cockroach cells is mainly discharged in the haemolymph in the form of potassium urate. Potassium urate is a crystalline substance that is insoluble in water. It is a relatively non-toxic and concentrated form of nitrogenous waste.

Reasons for Potassium Urate Discharge:
1. Solubility: Potassium urate is insoluble in water, which allows it to be more easily transported and stored in the haemolymph without causing harm to the cockroach's cells.
2. Concentration: Potassium urate is a concentrated form of nitrogenous waste, allowing the cockroach to conserve water by excreting waste in a form that requires minimal water loss.
3. Low Toxicity: Potassium urate is relatively non-toxic compared to other nitrogenous waste products like ammonia. This reduces the risk of cellular damage and allows the cockroach to safely store and eliminate waste without harming its own tissues.

Alternative Options:
a) Urea: Urea is a common nitrogenous waste product in mammals but is not the primary form of waste excretion in cockroaches.
c) Calcium Carbonate: Calcium carbonate is not a nitrogenous waste product and is primarily involved in the formation of the cockroach's exoskeleton.
d) Ammonia: Ammonia is a highly toxic nitrogenous waste product that requires large amounts of water for excretion. Cockroaches primarily excrete potassium urate to conserve water.

Conclusion:
In conclusion, cockroaches discharge their nitrogenous waste in the haemolymph mainly in the form of potassium urate. This form of waste excretion allows the cockroach to conserve water, minimize toxicity, and safely eliminate waste without harming its own cells.