Q.(a) Define temperature.

Temperature = the quantity that decides direction of heat flow; the zeroth law says mutual thermal equilibrium with a common third body makes two bodies equilibrium with each other (this is what makes 'temperature' meaningful/measurable); the second law forbids a heat engine of 100% efficiency (or heat flowing spontaneously from cold to hot).

1. Temperature: Temperature is a physical quantity that measures the degree of hotness or coldness of a body. More precisely, it is the quantity that determines the direction of net heat flow when two bodies are placed in thermal contact — heat always flows spontaneously from the body at higher temperature to the body at lower temperature, until both reach the same temperature (thermal equilibrium). Microscopically, temperature is related to the average kinetic energy of the molecules of the substance.
2. Zeroth law of thermodynamics — statement and proof/explanation: If two systems A and B are each separately in thermal equilibrium with a third system C, then A and B are also in thermal equilibrium with each other. Consider three systems A, B, C. Let A and C be separated by a diathermic (heat-conducting) wall and allowed to reach thermal equilibrium; likewise let B and C be separately brought to thermal equilibrium with C (say, A and B are both, in turn, placed in contact with the same body C, which acts like a thermometer). The zeroth law asserts that if A and B are now brought into contact through a diathermic wall, no further heat flows between them — they are already found to be in equilibrium. This shows that 'being in thermal equilibrium with each other' is a transitive relation, which is only possible if there exists a common physical quantity — temperature — that is equal for all systems in mutual thermal equilibrium. This is precisely the logical basis on which thermometers work: the thermometer plays the role of the common system C, and two bodies are said to be at the same temperature if each is separately found to be in equilibrium with the same thermometer reading. …