ANS:E - all the above.

1. Soil is homogeneous and fully saturated: Terzaghi's theory of one-dimensional consolidation assumes that the soil through which water is being squeezed out is homogeneous, meaning it has uniform properties throughout the depth being analyzed. Additionally, it assumes that the soil is fully saturated, meaning all void spaces between soil particles are filled with water.
2. Water and soil particles are incompressible: This assumption is common in soil mechanics and geotechnical engineering analyses. It implies that the volume of water and soil particles remains constant regardless of the applied load or pressure. While it simplifies calculations, it may not hold true in all situations, especially under high pressures.
3. Deformation of the soil is entirely due to a change in volume: In Terzaghi's theory, the primary mechanism of consolidation is the expulsion of water from the void spaces between soil particles. As water is squeezed out, the volume of void spaces decreases, leading to consolidation or settlement of the soil mass. While changes in volume are a key aspect, other factors such as rearrangement of soil particles and expulsion of air may also contribute to deformation.
4. Darcy's law for the velocity of flow of water through soil is perfectly valid: Darcy's law describes the flow of water through porous media like soil and is a fundamental principle in groundwater flow analysis. While it's relevant to the study of soil mechanics and consolidation, it's not specifically an assumption of Terzaghi's theory. However, it provides valuable insights into how water moves through saturated soil during the consolidation process.